Cubic critical

Percentage Accurate: 52.0% → 87.7%

Time: 8.3s

Alternatives: 18

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: 52.0% 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: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{a}{b} \cdot -1.5, \frac{c}{b}, 2\right)}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot -3, a, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375}{b}, a \cdot \frac{c}{b}, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+146)
   (/ (* (- b) (fma (* (/ a b) -1.5) (/ c b) 2.0)) (* 3.0 a))
   (if (<= b 1.25e-134)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (if (<= b 1.52e+25)
       (/
        (/ (fma (* c -3.0) a 0.0) (+ (sqrt (fma (* c -3.0) a (* b b))) b))
        (* 3.0 a))
       (/ (* (fma (/ -0.375 b) (* a (/ c b)) -0.5) c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+146) {
		tmp = (-b * fma(((a / b) * -1.5), (c / b), 2.0)) / (3.0 * a);
	} else if (b <= 1.25e-134) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else if (b <= 1.52e+25) {
		tmp = (fma((c * -3.0), a, 0.0) / (sqrt(fma((c * -3.0), a, (b * b))) + b)) / (3.0 * a);
	} else {
		tmp = (fma((-0.375 / b), (a * (c / b)), -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+146)
		tmp = Float64(Float64(Float64(-b) * fma(Float64(Float64(a / b) * -1.5), Float64(c / b), 2.0)) / Float64(3.0 * a));
	elseif (b <= 1.25e-134)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	elseif (b <= 1.52e+25)
		tmp = Float64(Float64(fma(Float64(c * -3.0), a, 0.0) / Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(-0.375 / b), Float64(a * Float64(c / b)), -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+146], N[(N[((-b) * N[(N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision] * N[(c / b), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-134], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.52e+25], N[(N[(N[(N[(c * -3.0), $MachinePrecision] * a + 0.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 / b), $MachinePrecision] * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.9999999999999999e146

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 49.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

   7. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 97.1

          \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{a}{b} \cdot -1.5, \color{blue}{\frac{c}{b}}, 2\right)}{3 \cdot a} \]

   9. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{a}{b} \cdot -1.5, \frac{c}{b}, 2\right)}}{3 \cdot a} \]

   if -4.9999999999999999e146 < b < 1.2500000000000001e-134

   1. Initial program 76.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 75.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.9

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

   6. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 76.0

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

   8. Applied rewrites 76.0%

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

   if 1.2500000000000001e-134 < b < 1.52000000000000006e25

   1. Initial program 39.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 39.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 39.8

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

   6. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 39.9

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

   8. Applied rewrites 39.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   10. Applied rewrites 86.8%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot -3, a, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{3 \cdot a} \]

   if 1.52000000000000006e25 < b

   1. Initial program 10.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{-1}{2} \cdot c}{b} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{b \cdot b}} + \frac{-1}{2} \cdot c}{b} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot {c}^{2}}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

      6. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot a}{b} \cdot \frac{{c}^{2}}{b}} + \frac{-1}{2} \cdot c}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{-3}{8} \cdot a}{b}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}}{b} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{a \cdot \frac{-3}{8}}}{b}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \frac{-3}{8}}{\color{blue}{b \cdot 1}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b} \cdot \frac{\frac{-3}{8}}{1}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \color{blue}{\frac{-3}{8}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b} \cdot \frac{-3}{8}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b}} \cdot \frac{-3}{8}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \color{blue}{\frac{{c}^{2}}{b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      17. lower-*.f64 72.6

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot -0.375, \frac{c \cdot c}{b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{b} \cdot -0.375, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.375}{b}, a \cdot \frac{c}{b}, -0.5\right) \cdot c}{b} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{a}{b} \cdot -1.5, \frac{c}{b}, 2\right)}{3 \cdot a}\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375}{b}, a \cdot \frac{c}{b}, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+146)
   (/ (* (- b) (fma (* (/ a b) -1.5) (/ c b) 2.0)) (* 3.0 a))
   (if (<= b 9.6e-88)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (/ (* (fma (/ -0.375 b) (* a (/ c b)) -0.5) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+146) {
		tmp = (-b * fma(((a / b) * -1.5), (c / b), 2.0)) / (3.0 * a);
	} else if (b <= 9.6e-88) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (fma((-0.375 / b), (a * (c / b)), -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+146)
		tmp = Float64(Float64(Float64(-b) * fma(Float64(Float64(a / b) * -1.5), Float64(c / b), 2.0)) / Float64(3.0 * a));
	elseif (b <= 9.6e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(-0.375 / b), Float64(a * Float64(c / b)), -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+146], N[(N[((-b) * N[(N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision] * N[(c / b), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.6e-88], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 / b), $MachinePrecision] * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999999e146

   1. Initial program 49.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 49.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

   7. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. associate-*r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 97.1

          \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{a}{b} \cdot -1.5, \color{blue}{\frac{c}{b}}, 2\right)}{3 \cdot a} \]

   9. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{a}{b} \cdot -1.5, \frac{c}{b}, 2\right)}}{3 \cdot a} \]

   if -4.9999999999999999e146 < b < 9.5999999999999998e-88

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 75.1

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

   8. Applied rewrites 75.1%

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

   if 9.5999999999999998e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{-1}{2} \cdot c}{b} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{b \cdot b}} + \frac{-1}{2} \cdot c}{b} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot {c}^{2}}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

      6. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot a}{b} \cdot \frac{{c}^{2}}{b}} + \frac{-1}{2} \cdot c}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{-3}{8} \cdot a}{b}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}}{b} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{a \cdot \frac{-3}{8}}}{b}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \frac{-3}{8}}{\color{blue}{b \cdot 1}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b} \cdot \frac{\frac{-3}{8}}{1}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \color{blue}{\frac{-3}{8}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b} \cdot \frac{-3}{8}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b}} \cdot \frac{-3}{8}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \color{blue}{\frac{{c}^{2}}{b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      17. lower-*.f64 71.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot -0.375, \frac{c \cdot c}{b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{b} \cdot -0.375, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.375}{b}, a \cdot \frac{c}{b}, -0.5\right) \cdot c}{b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, -0.5, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375}{b}, a \cdot \frac{c}{b}, -0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+116)
   (* (- b) (fma (/ (/ c b) b) -0.5 (/ 0.6666666666666666 a)))
   (if (<= b 9.6e-88)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (/ (* (fma (/ -0.375 b) (* a (/ c b)) -0.5) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+116) {
		tmp = -b * fma(((c / b) / b), -0.5, (0.6666666666666666 / a));
	} else if (b <= 9.6e-88) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (fma((-0.375 / b), (a * (c / b)), -0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+116)
		tmp = Float64(Float64(-b) * fma(Float64(Float64(c / b) / b), -0.5, Float64(0.6666666666666666 / a)));
	elseif (b <= 9.6e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(-0.375 / b), Float64(a * Float64(c / b)), -0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+116], N[((-b) * N[(N[(N[(c / b), $MachinePrecision] / b), $MachinePrecision] * -0.5 + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.6e-88], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 / b), $MachinePrecision] * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000002e116

   1. Initial program 50.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -1.00000000000000002e116 < b < 9.5999999999999998e-88

   1. Initial program 75.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 75.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.2

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

   6. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 75.3

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

   8. Applied rewrites 75.3%

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

   if 9.5999999999999998e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{-1}{2} \cdot c}{b} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{b \cdot b}} + \frac{-1}{2} \cdot c}{b} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot {c}^{2}}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

      6. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot a}{b} \cdot \frac{{c}^{2}}{b}} + \frac{-1}{2} \cdot c}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{-3}{8} \cdot a}{b}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}}{b} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{a \cdot \frac{-3}{8}}}{b}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot \frac{-3}{8}}{\color{blue}{b \cdot 1}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. times-frac N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b} \cdot \frac{\frac{-3}{8}}{1}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \color{blue}{\frac{-3}{8}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b} \cdot \frac{-3}{8}}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{a}{b}} \cdot \frac{-3}{8}, \frac{{c}^{2}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \color{blue}{\frac{{c}^{2}}{b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot \frac{-3}{8}, \frac{\color{blue}{c \cdot c}}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      17. lower-*.f64 71.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{a}{b} \cdot -0.375, \frac{c \cdot c}{b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

   5. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{b} \cdot -0.375, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.375}{b}, a \cdot \frac{c}{b}, -0.5\right) \cdot c}{b} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\left(-b\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, -0.5, \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+116)
   (* (- b) (fma (/ (/ c b) b) -0.5 (/ 0.6666666666666666 a)))
   (if (<= b 7.5e-88)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+116) {
		tmp = -b * fma(((c / b) / b), -0.5, (0.6666666666666666 / a));
	} else if (b <= 7.5e-88) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+116)
		tmp = Float64(Float64(-b) * fma(Float64(Float64(c / b) / b), -0.5, Float64(0.6666666666666666 / a)));
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+116], N[((-b) * N[(N[(N[(c / b), $MachinePrecision] / b), $MachinePrecision] * -0.5 + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000002e116

   1. Initial program 50.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -1.00000000000000002e116 < b < 7.50000000000000041e-88

   1. Initial program 75.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 75.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.2

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

   6. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 75.3

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

   8. Applied rewrites 75.3%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 5: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e+151)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e+151) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e+151)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e+151], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e151

   1. Initial program 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.9%

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

   if -3.4e151 < b < 7.50000000000000041e-88

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 75.1

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

   8. Applied rewrites 75.1%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 6: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e+151)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e+151) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e+151)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e+151], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e151

   1. Initial program 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.9%

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

   if -3.4e151 < b < 7.50000000000000041e-88

   1. Initial program 75.1%

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

   4. Applied rewrites 75.0%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 7: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e+151)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (* (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e+151) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = ((sqrt(fma((a * -3.0), c, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e+151)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e+151], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e151

   1. Initial program 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.9%

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

   if -3.4e151 < b < 7.50000000000000041e-88

   1. Initial program 75.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 74.9%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 8: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e+151)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e+151) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e+151)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e+151], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e151

   1. Initial program 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.9%

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

   if -3.4e151 < b < 7.50000000000000041e-88

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.9%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 9: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+146)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+146) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+146)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+146], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999999e146

   1. Initial program 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.9%

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

   if -4.9999999999999999e146 < b < 7.50000000000000041e-88

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 74.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 74.9

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

   4. Applied rewrites 74.8%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 10: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -3\right) \cdot a} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-110)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (/ (- (sqrt (* (* c -3.0) a)) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-110) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = (sqrt(((c * -3.0) * a)) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-110)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * -3.0) * a)) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-110], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999998e-110

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 83.6%

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

   if -3.7999999999999998e-110 < b < 7.50000000000000041e-88

   1. Initial program 64.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 61.8

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

   7. Applied rewrites 61.9%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 11: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-110)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-110) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-110)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-110], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999998e-110

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 83.6%

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

   if -3.7999999999999998e-110 < b < 7.50000000000000041e-88

   1. Initial program 64.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. lift--.f64 64.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 64.4

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 61.8

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

   9. Applied rewrites 61.8%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 12: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -3\right) \cdot a} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-110)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (* (/ (- (sqrt (* (* c -3.0) a)) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-110) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = ((sqrt(((c * -3.0) * a)) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-110)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * -3.0) * a)) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-110], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999998e-110

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 83.6%

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

   if -3.7999999999999998e-110 < b < 7.50000000000000041e-88

   1. Initial program 64.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 61.7%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 13: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\left(c \cdot -3\right) \cdot a} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-110)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (if (<= b 7.5e-88)
     (* (/ 0.3333333333333333 a) (- (sqrt (* (* c -3.0) a)) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-110) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else if (b <= 7.5e-88) {
		tmp = (0.3333333333333333 / a) * (sqrt(((c * -3.0) * a)) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-110)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	elseif (b <= 7.5e-88)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(Float64(c * -3.0) * a)) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-110], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999998e-110

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 83.6%

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

   if -3.7999999999999998e-110 < b < 7.50000000000000041e-88

   1. Initial program 64.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 61.6

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{a}} \cdot \left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{a} \cdot \color{blue}{\left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{1}{3}}{a} \cdot \color{blue}{\left(\sqrt{-3 \cdot \left(c \cdot a\right)} + \left(-b\right)\right)} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{1}{3}}{a} \cdot \left(\sqrt{-3 \cdot \left(c \cdot a\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      14. unsub-neg N/A

          \[\leadsto \frac{\frac{1}{3}}{a} \cdot \color{blue}{\left(\sqrt{-3 \cdot \left(c \cdot a\right)} - b\right)} \]

      15. lower--.f64 61.6

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(\sqrt{-3 \cdot \left(c \cdot a\right)} - b\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\left(c \cdot -3\right) \cdot a} - b\right)} \]

   if 7.50000000000000041e-88 < b

   1. Initial program 17.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 86.9

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (/ (fma -0.6666666666666666 b (* 0.5 (* a (/ c b)))) a)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = fma(-0.6666666666666666, b, (0.5 * (a * (c / b)))) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(a * Float64(c / b)))) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{3 \cdot a}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]

   4. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(\frac{2}{3} + \frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)}{a} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(-b\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{2}{3}\right)}}{a} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(-b\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)} + \frac{2}{3}\right)}{a} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(-b\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot a\right) \cdot \frac{c}{{b}^{2}}} + \frac{2}{3}\right)}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot a, \frac{c}{{b}^{2}}, \frac{2}{3}\right)}}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{2} \cdot a}, \frac{c}{{b}^{2}}, \frac{2}{3}\right)}{a} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot a, \color{blue}{\frac{c}{{b}^{2}}}, \frac{2}{3}\right)}{a} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot a, \frac{c}{\color{blue}{b \cdot b}}, \frac{2}{3}\right)}{a} \]

      12. lower-*.f64 63.9

          \[\leadsto \frac{\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot a, \frac{c}{\color{blue}{b \cdot b}}, 0.6666666666666666\right)}{a} \]

   7. Applied rewrites 63.9%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot a, \frac{c}{b \cdot b}, 0.6666666666666666\right)}}{a} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-2}{3} \cdot b + \color{blue}{\frac{1}{2} \cdot \frac{a \cdot c}{b}}}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.6666666666666666, \color{blue}{b}, 0.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a} \]

   if -9.999999999999969e-311 < b

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 68.4

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (* -2.0 b) (* 3.0 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-2.0 * b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (-2.0 * b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. lower-*.f64 65.4

         \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

   5. Applied rewrites 65.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

   if -9.999999999999969e-311 < b

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 68.4

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.6666666666666666}{a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* (/ -0.6666666666666666 a) b) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 / a) * b;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = ((-0.6666666666666666d0) / a) * b
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 / a) * b;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-0.6666666666666666 / a) * b
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-0.6666666666666666 / a) * b);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (-0.6666666666666666 / a) * b;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-0.6666666666666666 / a), $MachinePrecision] * b), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 65.2

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{b}}{-0.6666666666666666}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.3%

      \[\leadsto \frac{-0.6666666666666666}{a} \cdot \color{blue}{b} \]

   if -9.999999999999969e-311 < b

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 68.4

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 65.2

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if -9.999999999999969e-311 < b

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. lower-/.f64 68.4

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / 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 = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

   2. lower-/.f64 36.5

      \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

5. Applied rewrites 36.5%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))