Cubic critical

Percentage Accurate: 52.8% → 86.0%

Time: 7.9s

Alternatives: 12

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: 52.8% 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: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+145)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 2.55e-61)
     (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+145) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 2.55e-61) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+145)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 2.55e-61)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+145], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.55e-61], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e145

   1. Initial program 34.3%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -2e145 < b < 2.54999999999999984e-61

   1. Initial program 85.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 85.8

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

   4. Applied rewrites 85.8%

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

   if 2.54999999999999984e-61 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+145)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 2.55e-61)
     (/ (- (sqrt (fma b b (* (* -3.0 c) a))) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+145) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 2.55e-61) {
		tmp = (sqrt(fma(b, b, ((-3.0 * c) * a))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+145)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 2.55e-61)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * c) * a))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+145], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.55e-61], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e145

   1. Initial program 34.3%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -2e145 < b < 2.54999999999999984e-61

   1. Initial program 85.8%

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

   4. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 85.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.7

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

   6. Applied rewrites 85.7%

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

   if 2.54999999999999984e-61 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e+85)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 2.55e-61)
     (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) (/ 0.3333333333333333 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e+85) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 2.55e-61) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e+85)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 2.55e-61)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e+85], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.55e-61], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000002e85

   1. Initial program 43.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -1.7000000000000002e85 < b < 2.54999999999999984e-61

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 84.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   if 2.54999999999999984e-61 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot c\right) \cdot a} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-63)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 4.3e-65)
     (/ (- (sqrt (* (* -3.0 c) a)) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = (sqrt(((-3.0 * c) * a)) - b) / (a * 3.0);
	} 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 <= (-1.9d-63)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 4.3d-65) then
        tmp = (sqrt((((-3.0d0) * c) * a)) - b) / (a * 3.0d0)
    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 <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = (Math.sqrt(((-3.0 * c) * a)) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-63:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 4.3e-65:
		tmp = (math.sqrt(((-3.0 * c) * a)) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-63)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 4.3e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(-3.0 * c) * a)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-63)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 4.3e-65)
		tmp = (sqrt(((-3.0 * c) * a)) - b) / (a * 3.0);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-63], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-65], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.90000000000000009e-63

   1. Initial program 58.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -1.90000000000000009e-63 < b < 4.30000000000000024e-65

   1. Initial program 80.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 73.4

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

   7. Applied rewrites 73.5%

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

   if 4.30000000000000024e-65 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot c\right) \cdot a} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-63)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 4.3e-65)
     (/ (- (sqrt (* (* c a) -3.0)) b) (* a 3.0))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} 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 <= (-1.9d-63)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 4.3d-65) then
        tmp = (sqrt(((c * a) * (-3.0d0))) - b) / (a * 3.0d0)
    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 <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = (Math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-63:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 4.3e-65:
		tmp = (math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-63)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 4.3e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-63)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 4.3e-65)
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-63], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-65], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.90000000000000009e-63

   1. Initial program 58.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -1.90000000000000009e-63 < b < 4.30000000000000024e-65

   1. Initial program 80.6%

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if 4.30000000000000024e-65 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot c\right) \cdot a} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-63)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 4.3e-65)
     (* (/ (- (sqrt (* (* -3.0 c) a)) b) a) 0.3333333333333333)
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = ((sqrt(((-3.0 * c) * a)) - b) / a) * 0.3333333333333333;
	} 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 <= (-1.9d-63)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 4.3d-65) then
        tmp = ((sqrt((((-3.0d0) * c) * a)) - b) / a) * 0.3333333333333333d0
    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 <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = ((Math.sqrt(((-3.0 * c) * a)) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-63:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 4.3e-65:
		tmp = ((math.sqrt(((-3.0 * c) * a)) - b) / a) * 0.3333333333333333
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-63)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 4.3e-65)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(-3.0 * c) * a)) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-63)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 4.3e-65)
		tmp = ((sqrt(((-3.0 * c) * a)) - b) / a) * 0.3333333333333333;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-63], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-65], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.90000000000000009e-63

   1. Initial program 58.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -1.90000000000000009e-63 < b < 4.30000000000000024e-65

   1. Initial program 80.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 73.3%

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

   if 4.30000000000000024e-65 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(-3 \cdot c\right) \cdot a} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-63)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 4.3e-65)
     (* (- (sqrt (* (* -3.0 c) a)) b) (/ 0.3333333333333333 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = (sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / 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 <= (-1.9d-63)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 4.3d-65) then
        tmp = (sqrt((((-3.0d0) * c) * a)) - b) * (0.3333333333333333d0 / 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 <= -1.9e-63) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 4.3e-65) {
		tmp = (Math.sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-63:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 4.3e-65:
		tmp = (math.sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-63)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 4.3e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(-3.0 * c) * a)) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-63)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 4.3e-65)
		tmp = (sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-63], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-65], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.90000000000000009e-63

   1. Initial program 58.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -1.90000000000000009e-63 < b < 4.30000000000000024e-65

   1. Initial program 80.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 73.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 73.4

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

   7. Applied rewrites 73.3%

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

   if 4.30000000000000024e-65 < b

   1. Initial program 15.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 88.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.26e-219) (/ (* -2.0 b) (* a 3.0)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.26e-219) {
		tmp = (-2.0 * b) / (a * 3.0);
	} 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 <= 1.26d-219) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    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 <= 1.26e-219) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.26e-219:
		tmp = (-2.0 * b) / (a * 3.0)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.26e-219)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.26e-219)
		tmp = (-2.0 * b) / (a * 3.0);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.26e-219], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26000000000000003e-219

   1. Initial program 67.7%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if 1.26000000000000003e-219 < b

   1. Initial program 26.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 75.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.26e-219) (/ (* -0.6666666666666666 b) a) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.26e-219) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.26d-219) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.26e-219) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.26e-219:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.26e-219)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.26e-219)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.26e-219], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26000000000000003e-219

   1. Initial program 67.7%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

   7. Applied rewrites 66.2%

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

   if 1.26000000000000003e-219 < b

   1. Initial program 26.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 75.5%

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

Alternative 10: 67.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.26e-219) (* (/ b a) -0.6666666666666666) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.26e-219) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 1.26d-219) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 1.26e-219) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.26e-219:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.26e-219)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.26e-219)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.26e-219], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26000000000000003e-219

   1. Initial program 67.7%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if 1.26000000000000003e-219 < b

   1. Initial program 26.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 75.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.26e-219) (* (/ b a) -0.6666666666666666) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.26e-219) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.26d-219) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.26e-219) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.26e-219:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.26e-219)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.26e-219)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.26e-219], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26000000000000003e-219

   1. Initial program 67.7%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if 1.26000000000000003e-219 < b

   1. Initial program 26.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-219}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \cdot -0.6666666666666666 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ b a) -0.6666666666666666))

C

double code(double a, double b, double c) {
	return (b / a) * -0.6666666666666666;
}

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 / a) * (-0.6666666666666666d0)
end function

Java

public static double code(double a, double b, double c) {
	return (b / a) * -0.6666666666666666;
}

Python

def code(a, b, c):
	return (b / a) * -0.6666666666666666

Julia

function code(a, b, c)
	return Float64(Float64(b / a) * -0.6666666666666666)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (b / a) * -0.6666666666666666;
end

Wolfram

code[a_, b_, c_] := N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 37.2

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

5. Applied rewrites 37.2%

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

6. Final simplification 37.2%

   \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))