Cubic critical

Percentage Accurate: 52.3% → 85.4%

Time: 10.7s

Alternatives: 15

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% 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: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+143)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6e-73)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+143) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6e-73) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+143)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6e-73)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+143], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6e-73], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999998e143

   1. Initial program 47.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 47.1

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

   4. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 47.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 47.1

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.3

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

   9. Applied rewrites 97.3%

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

   if -8.4999999999999998e143 < b < 6e-73

   1. Initial program 79.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 79.1

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

   4. Applied rewrites 79.1%

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

   if 6e-73 < b

   1. Initial program 15.5%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+143)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6e-73)
     (/ (* (- (sqrt (fma c (* a -3.0) (* b b))) b) 0.3333333333333333) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+143) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6e-73) {
		tmp = ((sqrt(fma(c, (a * -3.0), (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+143)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6e-73)
		tmp = Float64(Float64(Float64(sqrt(fma(c, Float64(a * -3.0), Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+143], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6e-73], N[(N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999998e143

   1. Initial program 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 44.8

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

   4. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 44.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 44.7

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

   6. Applied rewrites 44.7%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.8

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

   9. Applied rewrites 96.8%

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

   if -8.4999999999999998e143 < b < 6e-73

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 81.7

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

   4. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if 6e-73 < b

   1. Initial program 18.8%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))