Cubic critical

Percentage Accurate: 52.9% → 85.5%

Time: 11.0s

Alternatives: 15

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-3}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+67)
   (/ (/ (- b (- b)) -3.0) a)
   (if (<= b 1.06e-54)
     (/ 1.0 (* a (/ -3.0 (- b (sqrt (fma b b (* c (* a -3.0))))))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+67) {
		tmp = ((b - -b) / -3.0) / a;
	} else if (b <= 1.06e-54) {
		tmp = 1.0 / (a * (-3.0 / (b - sqrt(fma(b, b, (c * (a * -3.0)))))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+67)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / -3.0) / a);
	elseif (b <= 1.06e-54)
		tmp = Float64(1.0 / Float64(a * Float64(-3.0 / Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+67], N[(N[(N[(b - (-b)), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(1.0 / N[(a * N[(-3.0 / N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000001e67

   1. Initial program 64.7%

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

   4. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 64.8

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 64.6

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   9. Applied rewrites 97.9%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       9. lower-/.f64 98.1

          \[\leadsto \frac{\color{blue}{\frac{b - \left(-b\right)}{-3}}}{a} \]

   11. Applied rewrites 98.1%

       \[\leadsto \color{blue}{\frac{\frac{b - \left(-b\right)}{-3}}{a}} \]

   if -5.2000000000000001e67 < b < 1.0600000000000001e-54

   1. Initial program 76.3%

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

   4. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 76.2

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 76.2

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right) \cdot \color{blue}{\frac{1}{a \cdot -3}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{a \cdot -3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{\color{blue}{a \cdot -3}} \]

      6. *-commutative N/A

         \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{\color{blue}{-3 \cdot a}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{-3}}{a}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{-3}}{a}} \]

   8. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}{-3}}{a}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}{-3}}{a}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}{-3} \cdot \frac{1}{a}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}{-3}} \cdot \frac{1}{a} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}} \cdot \frac{1}{a} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}} \cdot a}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}} \cdot a} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}} \cdot a}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}} \cdot a}} \]

   10. Applied rewrites 76.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)}} \cdot a}} \]

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-3}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;\left(b - \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e+101)
   (* (- b (- b)) (/ (/ 1.0 a) -3.0))
   (if (<= b 1.06e-54)
     (/ (/ (- b (sqrt (fma a (* -3.0 c) (* b b)))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+101) {
		tmp = (b - -b) * ((1.0 / a) / -3.0);
	} else if (b <= 1.06e-54) {
		tmp = ((b - sqrt(fma(a, (-3.0 * c), (b * b)))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e+101)
		tmp = Float64(Float64(b - Float64(-b)) * Float64(Float64(1.0 / a) / -3.0));
	elseif (b <= 1.06e-54)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e+101], N[(N[(b - (-b)), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.6000000000000003e101

   1. Initial program 62.8%

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

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 62.9

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 62.7

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   9. Applied rewrites 97.9%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 98.0

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

   11. Applied rewrites 98.0%

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

   if -4.6000000000000003e101 < b < 1.0600000000000001e-54

   1. Initial program 77.0%

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

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;\left(b - \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+67)
   (/ (/ (- b (- b)) -3.0) a)
   (if (<= b 1.06e-54)
     (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+67) {
		tmp = ((b - -b) / -3.0) / a;
	} else if (b <= 1.06e-54) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+67)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / -3.0) / a);
	elseif (b <= 1.06e-54)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+67], N[(N[(N[(b - (-b)), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000001e67

   1. Initial program 64.7%

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

   4. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 64.8

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 64.6

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   9. Applied rewrites 97.9%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       9. lower-/.f64 98.1

          \[\leadsto \frac{\color{blue}{\frac{b - \left(-b\right)}{-3}}}{a} \]

   11. Applied rewrites 98.1%

       \[\leadsto \color{blue}{\frac{\frac{b - \left(-b\right)}{-3}}{a}} \]

   if -5.2000000000000001e67 < b < 1.0600000000000001e-54

   1. Initial program 76.3%

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

   4. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 76.2

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 76.2

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right) \cdot \color{blue}{\frac{1}{a \cdot -3}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{a \cdot -3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{\color{blue}{a \cdot -3}} \]

      6. *-commutative N/A

         \[\leadsto \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{\color{blue}{-3 \cdot a}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{-3}}{a}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}{-3}}{a}} \]

   8. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}{-3}}{a}} \]

   10. Applied rewrites 76.3%

       \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}} \]

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+67)
   (/ (/ (- b (- b)) -3.0) a)
   (if (<= b 1.06e-54)
     (/ (- (sqrt (fma a (* -3.0 c) (* b b))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+67) {
		tmp = ((b - -b) / -3.0) / a;
	} else if (b <= 1.06e-54) {
		tmp = (sqrt(fma(a, (-3.0 * c), (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+67)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / -3.0) / a);
	elseif (b <= 1.06e-54)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+67], N[(N[(N[(b - (-b)), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000001e67

   1. Initial program 64.7%

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

   4. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 64.8

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 64.6

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   9. Applied rewrites 97.9%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       9. lower-/.f64 98.1

          \[\leadsto \frac{\color{blue}{\frac{b - \left(-b\right)}{-3}}}{a} \]

   11. Applied rewrites 98.1%

       \[\leadsto \color{blue}{\frac{\frac{b - \left(-b\right)}{-3}}{a}} \]

   if -5.2000000000000001e67 < b < 1.0600000000000001e-54

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      5. lower--.f64 76.3

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(a, \left(\mathsf{neg}\left(3\right)\right) \cdot c, b \cdot b\right)}} - b}{3 \cdot a} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot c}, b \cdot b\right)} - b}{3 \cdot a} \]

      17. metadata-eval 76.3

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, \color{blue}{-3} \cdot c, b \cdot b\right)} - b}{3 \cdot a} \]

   4. Applied rewrites 76.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}}{3 \cdot a} \]

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{+100}:\\ \;\;\;\;\left(b - \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.26e+100)
   (* (- b (- b)) (/ (/ 1.0 a) -3.0))
   (if (<= b 1.06e-54)
     (* (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) a) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e+100) {
		tmp = (b - -b) * ((1.0 / a) / -3.0);
	} else if (b <= 1.06e-54) {
		tmp = ((sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.26e+100)
		tmp = Float64(Float64(b - Float64(-b)) * Float64(Float64(1.0 / a) / -3.0));
	elseif (b <= 1.06e-54)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.26e+100], N[(N[(b - (-b)), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2599999999999999e100

   1. Initial program 62.8%

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

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 62.9

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 62.7

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   9. Applied rewrites 97.9%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 98.0

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

   11. Applied rewrites 98.0%

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

   if -1.2599999999999999e100 < b < 1.0600000000000001e-54

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, b \cdot b\right)}}}{3 \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right), c, b \cdot b\right)}}{3 \cdot a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right), c, b \cdot b\right)}}{3 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      11. metadata-eval 77.0

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot \color{blue}{-3}, c, b \cdot b\right)}}{3 \cdot a} \]

   4. Applied rewrites 77.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{3 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{3 \cdot a}} \]

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a} \cdot 0.3333333333333333} \]

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{+100}:\\ \;\;\;\;\left(b - \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.85e+66)
   (/ (/ (- b (- b)) -3.0) a)
   (if (<= b 1.06e-54)
     (* (- b (sqrt (fma a (* -3.0 c) (* b b)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.85e+66) {
		tmp = ((b - -b) / -3.0) / a;
	} else if (b <= 1.06e-54) {
		tmp = (b - sqrt(fma(a, (-3.0 * c), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.85e+66)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / -3.0) / a);
	elseif (b <= 1.06e-54)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.85e+66], N[(N[(N[(b - (-b)), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8500000000000002e66

   1. Initial program 65.3%

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

   4. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 65.4

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 65.2

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   9. Applied rewrites 98.0%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       9. lower-/.f64 98.1

          \[\leadsto \frac{\color{blue}{\frac{b - \left(-b\right)}{-3}}}{a} \]

   11. Applied rewrites 98.1%

       \[\leadsto \color{blue}{\frac{\frac{b - \left(-b\right)}{-3}}{a}} \]

   if -2.8500000000000002e66 < b < 1.0600000000000001e-54

   1. Initial program 76.1%

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

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-33)
   (fma 0.5 (/ c b) (/ (* b -0.6666666666666666) a))
   (if (<= b 1.55e-52)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-33) {
		tmp = fma(0.5, (c / b), ((b * -0.6666666666666666) / a));
	} else if (b <= 1.55e-52) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-33)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b * -0.6666666666666666) / a));
	elseif (b <= 1.55e-52)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-33], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-52], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2e-33

   1. Initial program 70.8%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. lower-neg.f64 89.8

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b}}, \frac{b \cdot -0.6666666666666666}{a}\right) \]

   if -8.2e-33 < b < 1.5499999999999999e-52

   1. Initial program 72.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, b \cdot b\right)}}}{3 \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right), c, b \cdot b\right)}}{3 \cdot a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right), c, b \cdot b\right)}}{3 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      11. metadata-eval 72.5

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot \color{blue}{-3}, c, b \cdot b\right)}}{3 \cdot a} \]

   4. Applied rewrites 72.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{3 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{3 \cdot a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{a \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{a}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{a}}{3}} \]

   6. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a}}{3}} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}} - b}{a}}{3} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}} - b}{a}}{3} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3} - b}{a}}{3} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} - b}{a}}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(-3 \cdot a\right)}} - b}{a}}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)}} - b}{a}}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -3\right)}} - b}{a}}{3} \]

      7. lower-*.f64 64.9

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -3\right)}} - b}{a}}{3} \]

   9. Applied rewrites 64.9%

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} - b}{a}}{3} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}{3}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}}{3} \]

       3. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]

       4. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\sqrt{c \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(-3 \cdot a\right)\right)} - b}{\color{blue}{a \cdot 3}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{c \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(-3 \cdot a\right)\right)} - b}{\color{blue}{a \cdot 3}} \]

   11. Applied rewrites 64.9%

       \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(-3 \cdot a\right)} - b}{a \cdot 3}} \]

   if 1.5499999999999999e-52 < b

   1. Initial program 23.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-33)
   (fma 0.5 (/ c b) (/ (* b -0.6666666666666666) a))
   (if (<= b 1.55e-52)
     (* 0.3333333333333333 (/ (- (sqrt (* c (* a -3.0))) b) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-33) {
		tmp = fma(0.5, (c / b), ((b * -0.6666666666666666) / a));
	} else if (b <= 1.55e-52) {
		tmp = 0.3333333333333333 * ((sqrt((c * (a * -3.0))) - b) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-33)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b * -0.6666666666666666) / a));
	elseif (b <= 1.55e-52)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-33], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-52], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2e-33

   1. Initial program 70.8%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. lower-neg.f64 89.8

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b}}, \frac{b \cdot -0.6666666666666666}{a}\right) \]

   if -8.2e-33 < b < 1.5499999999999999e-52

   1. Initial program 72.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, b \cdot b\right)}}}{3 \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right), c, b \cdot b\right)}}{3 \cdot a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right), c, b \cdot b\right)}}{3 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      11. metadata-eval 72.5

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot \color{blue}{-3}, c, b \cdot b\right)}}{3 \cdot a} \]

   4. Applied rewrites 72.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{3 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{3 \cdot a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{a \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{a}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{a}}{3}} \]

   6. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a}}{3}} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}} - b}{a}}{3} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}} - b}{a}}{3} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3} - b}{a}}{3} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} - b}{a}}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(-3 \cdot a\right)}} - b}{a}}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)}} - b}{a}}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -3\right)}} - b}{a}}{3} \]

      7. lower-*.f64 64.9

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -3\right)}} - b}{a}}{3} \]

   9. Applied rewrites 64.9%

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} - b}{a}}{3} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}{3}} \]

       2. div-inv N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 64.9

          \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a} \cdot 0.3333333333333333} \]

   11. Applied rewrites 64.9%

       \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(-3 \cdot a\right)} - b}{a} \cdot 0.3333333333333333} \]

   if 1.5499999999999999e-52 < b

   1. Initial program 23.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-33)
   (fma 0.5 (/ c b) (/ (* b -0.6666666666666666) a))
   (if (<= b 1.55e-52)
     (* (- (sqrt (* c (* a -3.0))) b) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-33) {
		tmp = fma(0.5, (c / b), ((b * -0.6666666666666666) / a));
	} else if (b <= 1.55e-52) {
		tmp = (sqrt((c * (a * -3.0))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-33)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b * -0.6666666666666666) / a));
	elseif (b <= 1.55e-52)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-33], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-52], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2e-33

   1. Initial program 70.8%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. lower-neg.f64 89.8

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b}}, \frac{b \cdot -0.6666666666666666}{a}\right) \]

   if -8.2e-33 < b < 1.5499999999999999e-52

   1. Initial program 72.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, b \cdot b\right)}}}{3 \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right), c, b \cdot b\right)}}{3 \cdot a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right), c, b \cdot b\right)}}{3 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot \left(\mathsf{neg}\left(3\right)\right)}, c, b \cdot b\right)}}{3 \cdot a} \]

      11. metadata-eval 72.5

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot \color{blue}{-3}, c, b \cdot b\right)}}{3 \cdot a} \]

   4. Applied rewrites 72.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{3 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{3 \cdot a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{\color{blue}{a \cdot 3}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{a}}{3}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)}}{a}}{3}} \]

   6. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a}}{3}} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}} - b}{a}}{3} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}} - b}{a}}{3} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3} - b}{a}}{3} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} - b}{a}}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(-3 \cdot a\right)}} - b}{a}}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)}} - b}{a}}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -3\right)}} - b}{a}}{3} \]

      7. lower-*.f64 64.9

         \[\leadsto \frac{\frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -3\right)}} - b}{a}}{3} \]

   9. Applied rewrites 64.9%

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} - b}{a}}{3} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}{3}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}}}{3} \]

       3. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]

       4. div-inv N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/r* N/A

          \[\leadsto \left(\sqrt{c \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(-3 \cdot a\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}} \]

       7. metadata-eval N/A

          \[\leadsto \left(\sqrt{c \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(-3 \cdot a\right)\right)} - b\right) \cdot \frac{\color{blue}{\frac{1}{3}}}{a} \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{c \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(-3 \cdot a\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}} \]

   11. Applied rewrites 64.8%

       \[\leadsto \color{blue}{\left(\sqrt{c \cdot \left(-3 \cdot a\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

   if 1.5499999999999999e-52 < b

   1. Initial program 23.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.8

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b \cdot -0.6666666666666666}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (fma 0.5 (/ c b) (/ (* b -0.6666666666666666) a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = fma(0.5, (c / b), ((b * -0.6666666666666666) / a));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b * -0.6666666666666666) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 73.9%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. lower-neg.f64 65.1

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.1%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b}}, \frac{b \cdot -0.6666666666666666}{a}\right) \]

   if -4.999999999999985e-310 < b

   1. Initial program 33.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 67.4

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-300) (/ (/ (- b (- b)) -3.0) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-300) {
		tmp = ((b - -b) / -3.0) / a;
	} else {
		tmp = (c * -0.5) / 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 <= 8d-300) then
        tmp = ((b - -b) / (-3.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-300) {
		tmp = ((b - -b) / -3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-300:
		tmp = ((b - -b) / -3.0) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-300)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / -3.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e-300)
		tmp = ((b - -b) / -3.0) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-300], N[(N[(N[(b - (-b)), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.0000000000000002e-300

   1. Initial program 74.1%

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

   4. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}{-3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{a}}}{-3} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{a}}}{-3} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      7. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{-3 \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(3 \cdot a\right)}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

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

          \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{-3} \cdot a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      14. lift-*.f64 74.1

          \[\leadsto \frac{1}{\color{blue}{a \cdot -3}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \]

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}}\right) \]

      19. lower-*.f64 74.0

          \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right)}\right) \]

   6. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)}\right)} \]

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{-1 \cdot b}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 65.2

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

   9. Applied rewrites 65.2%

      \[\leadsto \frac{1}{a \cdot -3} \cdot \left(b - \color{blue}{\left(-b\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{b - \left(\mathsf{neg}\left(b\right)\right)}{-3}}{a}} \]

       9. lower-/.f64 65.4

          \[\leadsto \frac{\color{blue}{\frac{b - \left(-b\right)}{-3}}}{a} \]

   11. Applied rewrites 65.4%

       \[\leadsto \color{blue}{\frac{\frac{b - \left(-b\right)}{-3}}{a}} \]

   if 8.0000000000000002e-300 < b

   1. Initial program 33.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 67.9

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 66.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-300) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-300) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / 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 <= 8d-300) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-300) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-300:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-300)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e-300)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-300], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.0000000000000002e-300

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 65.3

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if 8.0000000000000002e-300 < b

   1. Initial program 33.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 67.9

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.8e+65) (/ (* b -0.6666666666666666) a) (/ (* c 0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e+65) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * 0.5) / 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+65) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * 0.5d0) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e+65) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * 0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e+65:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * 0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e+65)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * 0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.8e+65)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * 0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e+65], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000001e65

   1. Initial program 65.4%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 43.4

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if 5.8000000000000001e65 < b

   1. Initial program 17.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. lower-neg.f64 2.3

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Applied rewrites 2.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 33.5%

      \[\leadsto \frac{c \cdot 0.5}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 42.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.8e+65) (* b (/ -0.6666666666666666 a)) (/ (* c 0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e+65) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c * 0.5) / 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+65) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c * 0.5d0) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e+65) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c * 0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e+65:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c * 0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e+65)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c * 0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.8e+65)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c * 0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e+65], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000001e65

   1. Initial program 65.4%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 43.4

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.4%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]

   if 5.8000000000000001e65 < b

   1. Initial program 17.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

      15. lower-neg.f64 2.3

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Applied rewrites 2.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 33.5%

      \[\leadsto \frac{c \cdot 0.5}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 10.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot 0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* c 0.5) b))

C

double code(double a, double b, double c) {
	return (c * 0.5) / 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 = (c * 0.5d0) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * 0.5) / b;
}

Python

def code(a, b, c):
	return (c * 0.5) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * 0.5) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * 0.5) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot b}\right) \]

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

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{c \cdot \frac{-1}{2}}}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \left(\color{blue}{c \cdot \frac{\frac{-1}{2}}{{b}^{2}}} + \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{-1}{2}}{{b}^{2}}, \frac{2}{3} \cdot \frac{1}{a}\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-1}{2}}{{b}^{2}}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{\color{blue}{b \cdot b}}, \frac{2}{3} \cdot \frac{1}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   14. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(c, \frac{\frac{-1}{2}}{b \cdot b}, \color{blue}{\frac{\frac{2}{3}}{a}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   15. lower-neg.f64 32.2

       \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

5. Applied rewrites 32.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

6. Taylor expanded in c around inf

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

8. Applied rewrites 10.8%

   \[\leadsto \frac{c \cdot 0.5}{\color{blue}{b}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))