Cubic critical

Percentage Accurate: 52.9% → 85.6%

Time: 11.2s

Alternatives: 15

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+87)
   (/ (/ b -1.5) a)
   (if (<= b 1.4e-29)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+87) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e+87)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e+87], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.74999999999999993e87

   1. Initial program 61.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.5%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if -1.74999999999999993e87 < b < 1.4000000000000001e-29

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 79.8

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lower-fma.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 79.8

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

   6. Applied rewrites 79.8%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+87)
   (/ (/ b -1.5) a)
   (if (<= b 1.4e-29)
     (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+87) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e+87)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e+87], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.74999999999999993e87

   1. Initial program 61.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.5%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if -1.74999999999999993e87 < b < 1.4000000000000001e-29

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 79.8

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 79.7

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

   4. Applied rewrites 79.7%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+102)
   (/ (/ b -1.5) a)
   (if (<= b 1.4e-29)
     (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+102) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+102)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+102], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999996e102

   1. Initial program 59.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.3%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if -8.4999999999999996e102 < b < 1.4000000000000001e-29

   1. Initial program 80.1%

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

   4. Applied rewrites 79.9%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e-74)
   (* (fma c (/ -0.5 (* b b)) (/ 0.6666666666666666 a)) (- b))
   (if (<= b 1.4e-29)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = fma(c, (-0.5 / (b * b)), (0.6666666666666666 / a)) * -b;
	} else if (b <= 1.4e-29) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e-74)
		tmp = Float64(fma(c, Float64(-0.5 / Float64(b * b)), Float64(0.6666666666666666 / a)) * Float64(-b));
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e-74], N[(N[(c * N[(-0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1600000000000001e-74

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -1.1600000000000001e-74 < b < 1.4000000000000001e-29

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 76.1

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

   4. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lower-fma.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.0

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

   9. Applied rewrites 73.0%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e-74)
   (/ (/ b -1.5) a)
   (if (<= b 1.4e-29)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.16d-74)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 1.4d-29) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.16e-74:
		tmp = (b / -1.5) / a
	elif b <= 1.4e-29:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e-74)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.16e-74)
		tmp = (b / -1.5) / a;
	elseif (b <= 1.4e-29)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e-74], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1600000000000001e-74

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if -1.1600000000000001e-74 < b < 1.4000000000000001e-29

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 76.1

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

   4. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lower-fma.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.0

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

   9. Applied rewrites 73.0%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

Alternative 6: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e-74)
   (/ (/ b -1.5) a)
   (if (<= b 1.4e-29)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.16d-74)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 1.4d-29) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.16e-74:
		tmp = (b / -1.5) / a
	elif b <= 1.4e-29:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e-74)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.16e-74)
		tmp = (b / -1.5) / a;
	elseif (b <= 1.4e-29)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e-74], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1600000000000001e-74

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if -1.1600000000000001e-74 < b < 1.4000000000000001e-29

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 76.3

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.8

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

   7. Applied rewrites 72.8%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e-74)
   (/ (/ b -1.5) a)
   (if (<= b 1.4e-29)
     (* (/ -0.3333333333333333 a) (- b (sqrt (* -3.0 (* a c)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt((-3.0 * (a * c))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.16d-74)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 1.4d-29) then
        tmp = ((-0.3333333333333333d0) / a) * (b - sqrt(((-3.0d0) * (a * c))))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-74) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.4e-29) {
		tmp = (-0.3333333333333333 / a) * (b - Math.sqrt((-3.0 * (a * c))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.16e-74:
		tmp = (b / -1.5) / a
	elif b <= 1.4e-29:
		tmp = (-0.3333333333333333 / a) * (b - math.sqrt((-3.0 * (a * c))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e-74)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.4e-29)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(Float64(-3.0 * Float64(a * c)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.16e-74)
		tmp = (b / -1.5) / a;
	elseif (b <= 1.4e-29)
		tmp = (-0.3333333333333333 / a) * (b - sqrt((-3.0 * (a * c))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e-74], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.4e-29], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1600000000000001e-74

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if -1.1600000000000001e-74 < b < 1.4000000000000001e-29

   1. Initial program 76.3%

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.7

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

   7. Applied rewrites 72.7%

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

   if 1.4000000000000001e-29 < b

   1. Initial program 8.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-288) (/ (/ b -1.5) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-288) then
        tmp = (b / (-1.5d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-288:
		tmp = (b / -1.5) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-288)
		tmp = Float64(Float64(b / -1.5) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-288)
		tmp = (b / -1.5) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-288], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999999e-288

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

      \[\leadsto \frac{\frac{b}{-1.5}}{a} \]

   if 2.99999999999999999e-288 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 9: 68.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-288) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-288) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-288:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-288)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-288)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-288], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999999e-288

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 2.99999999999999999e-288 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 10: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-288) (/ (* b -0.6666666666666666) a) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-288) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-288:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-288)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-288)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-288], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999999e-288

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 2.99999999999999999e-288 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-288) (/ b (* a -1.5)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = b / (a * -1.5);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-288) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = b / (a * -1.5);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-288:
		tmp = b / (a * -1.5)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-288)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-288)
		tmp = b / (a * -1.5);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-288], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999999e-288

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.7%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]

   if 2.99999999999999999e-288 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-288) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-288) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-288:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-288)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-288)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-288], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999999e-288

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in b around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 2.99999999999999999e-288 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-288) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-288) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-288) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-288:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-288)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-288)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-288], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999999e-288

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.6%

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

   if 2.99999999999999999e-288 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-288}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.6e-9) (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.6e-9) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8.6d-9) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.6e-9) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.6e-9:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.6e-9)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.6e-9)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.6e-9], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.59999999999999925e-9

   1. Initial program 71.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 48.6

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

   5. Applied rewrites 48.6%

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

   7. Applied rewrites 48.5%

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

   if 8.59999999999999925e-9 < b

   1. Initial program 9.0%

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

   4. Applied rewrites 4.9%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 25.9

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

   7. Applied rewrites 25.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 11.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* 0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.5d0 * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 * (c / b);
}

Python

def code(a, b, c):
	return 0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

4. Applied rewrites 34.9%

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

5. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 10.3

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

7. Applied rewrites 10.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))