Cubic critical

Percentage Accurate: 51.3% → 85.6%

Time: 7.7s

Alternatives: 13

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: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a \cdot \frac{c}{b}}{b}, -1.5, 2\right) \cdot \left(-b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-92}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+104)
   (/ (* (fma (/ (* a (/ c b)) b) -1.5 2.0) (- b)) (* 3.0 a))
   (if (<= b 3e-92)
     (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+104) {
		tmp = (fma(((a * (c / b)) / b), -1.5, 2.0) * -b) / (3.0 * a);
	} else if (b <= 3e-92) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+104)
		tmp = Float64(Float64(fma(Float64(Float64(a * Float64(c / b)) / b), -1.5, 2.0) * Float64(-b)) / Float64(3.0 * a));
	elseif (b <= 3e-92)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+104], N[(N[(N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * -1.5 + 2.0), $MachinePrecision] * (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-92], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999997e104

   1. Initial program 59.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -4.9999999999999997e104 < b < 3.00000000000000013e-92

   1. Initial program 78.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.1

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

   4. Applied rewrites 78.1%

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

   if 3.00000000000000013e-92 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

Alternative 2: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-92}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.7e+103)
   (/ (fma 0.5 (* a (/ c b)) (* -0.6666666666666666 b)) a)
   (if (<= b 3e-92)
     (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.7e+103) {
		tmp = fma(0.5, (a * (c / b)), (-0.6666666666666666 * b)) / a;
	} else if (b <= 3e-92) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.7e+103)
		tmp = Float64(fma(0.5, Float64(a * Float64(c / b)), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 3e-92)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.7e+103], N[(N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3e-92], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.70000000000000033e103

   1. Initial program 59.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.3%

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

   if -4.70000000000000033e103 < b < 3.00000000000000013e-92

   1. Initial program 78.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.1

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

   4. Applied rewrites 78.1%

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

   if 3.00000000000000013e-92 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-92}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+103)
   (/ (fma 0.5 (* a (/ c b)) (* -0.6666666666666666 b)) a)
   (if (<= b 3e-92)
     (* 0.3333333333333333 (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+103) {
		tmp = fma(0.5, (a * (c / b)), (-0.6666666666666666 * b)) / a;
	} else if (b <= 3e-92) {
		tmp = 0.3333333333333333 * ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+103)
		tmp = Float64(fma(0.5, Float64(a * Float64(c / b)), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 3e-92)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+103], N[(N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3e-92], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.60000000000000017e103

   1. Initial program 59.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.3%

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

   if -3.60000000000000017e103 < b < 3.00000000000000013e-92

   1. Initial program 78.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.1

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. div-add N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. metadata-eval 78.0

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

   8. Applied rewrites 78.0%

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

   10. Applied rewrites 77.8%

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

   if 3.00000000000000013e-92 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-92}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-92}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot a\right) \cdot -3}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-83)
   (/ (fma 0.5 (* a (/ c b)) (* -0.6666666666666666 b)) a)
   (if (<= b 3e-92)
     (/ (+ (- b) (sqrt (* (* c a) -3.0))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-83) {
		tmp = fma(0.5, (a * (c / b)), (-0.6666666666666666 * b)) / a;
	} else if (b <= 3e-92) {
		tmp = (-b + sqrt(((c * a) * -3.0))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-83)
		tmp = Float64(fma(0.5, Float64(a * Float64(c / b)), Float64(-0.6666666666666666 * b)) / a);
	elseif (b <= 3e-92)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(c * a) * -3.0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-83], N[(N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3e-92], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2999999999999999e-83

   1. Initial program 72.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.2%

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

   if -2.2999999999999999e-83 < b < 3.00000000000000013e-92

   1. Initial program 73.5%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if 3.00000000000000013e-92 < b

   1. Initial program 17.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

Alternative 5: 68.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 70.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 6: 68.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 70.0%

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

   9. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{1}{2} \cdot \frac{a \cdot c}{b}}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 3.6%

       \[\leadsto \frac{0.5 \cdot \left(a \cdot \frac{c}{b}\right)}{a} \]

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 70.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 7: 68.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 70.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 8: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6e-308) (/ (/ (* -2.0 b) 3.0) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 1.6d-308) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e-308:
		tmp = ((-2.0 * b) / 3.0) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e-308)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.6e-308)
		tmp = ((-2.0 * b) / 3.0) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e-308], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001e-308

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 1.6000000000000001e-308 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 9: 68.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6e-308) (/ (* -2.0 b) (* 3.0 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 1.6d-308) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e-308:
		tmp = (-2.0 * b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e-308)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.6e-308)
		tmp = (-2.0 * b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e-308], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001e-308

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if 1.6000000000000001e-308 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 10: 68.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6e-308) (/ (* -0.6666666666666666 b) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 1.6d-308) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e-308:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e-308)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.6e-308)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e-308], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001e-308

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.2%

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

   if 1.6000000000000001e-308 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 11: 68.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6e-308) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 1.6d-308) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e-308:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e-308)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.6e-308)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e-308], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001e-308

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if 1.6000000000000001e-308 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 12: 44.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* -0.6666666666666666 (/ b a)) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 30.8

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

   4. Applied rewrites 30.8%

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

   6. Applied rewrites 30.8%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. div-add N/A

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

      5. frac-add N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. metadata-eval 28.1

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

   8. Applied rewrites 28.1%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{9} \cdot \left(-3 \cdot b + 3 \cdot b\right)}}{a} \]
   10. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{\left(-3 \cdot b - \left(\mathsf{neg}\left(3\right)\right) \cdot b\right)}}{a} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{9} \cdot \left(-3 \cdot b - \color{blue}{-3} \cdot b\right)}{a} \]

       3. +-inverses N/A

          \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{0}}{a} \]

       4. metadata-eval 17.8

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

   11. Applied rewrites 17.8%

       \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 11.2% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 49.7

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

4. Applied rewrites 49.7%

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

6. Applied rewrites 49.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. div-add N/A

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

   5. frac-add N/A

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

   6. lower-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. metadata-eval 48.2

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

8. Applied rewrites 48.2%

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

9. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{9} \cdot \left(-3 \cdot b + 3 \cdot b\right)}}{a} \]
10. Step-by-step derivation

    1. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{\left(-3 \cdot b - \left(\mathsf{neg}\left(3\right)\right) \cdot b\right)}}{a} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{9} \cdot \left(-3 \cdot b - \color{blue}{-3} \cdot b\right)}{a} \]

    3. +-inverses N/A

       \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{0}}{a} \]

    4. metadata-eval 11.2

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

11. Applied rewrites 11.2%

    \[\leadsto \frac{\color{blue}{0}}{a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))