Cubic critical

Percentage Accurate: 52.5% → 85.3%

Time: 8.8s

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: 52.5% 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: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+135)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 7.2e-75)
     (/ (/ (- (sqrt (fma a (* -3.0 c) (* b b))) b) 3.0) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+135) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 7.2e-75) {
		tmp = ((sqrt(fma(a, (-3.0 * c), (b * b))) - b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+135)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 7.2e-75)
		tmp = Float64(Float64(Float64(sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))) - b) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+135], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-75], N[(N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1999999999999999e135

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 35.6

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 35.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.6

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

   6. Applied rewrites 35.6%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if -2.1999999999999999e135 < b < 7.2000000000000001e-75

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 85.2

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 85.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 85.2

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

   6. Applied rewrites 85.2%

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

   if 7.2000000000000001e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e+135)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 7.2e-75)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+135) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 7.2e-75) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d+135)) then
        tmp = ((-b - b) / 3.0d0) / a
    else if (b <= 7.2d-75) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+135) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 7.2e-75) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e+135:
		tmp = ((-b - b) / 3.0) / a
	elif b <= 7.2e-75:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e+135)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 7.2e-75)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e+135)
		tmp = ((-b - b) / 3.0) / a;
	elseif (b <= 7.2e-75)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e+135], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-75], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000001e135

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 35.6

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 35.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.6

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

   6. Applied rewrites 35.6%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if -2.1000000000000001e135 < b < 7.2000000000000001e-75

   1. Initial program 85.1%

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

   if 7.2000000000000001e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e+135)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 7.2e-75)
     (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+135) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 7.2e-75) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e+135)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 7.2e-75)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e+135], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-75], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000001e135

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 35.6

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 35.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.6

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

   6. Applied rewrites 35.6%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if -2.1000000000000001e135 < b < 7.2000000000000001e-75

   1. Initial program 85.1%

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

   4. Applied rewrites 85.1%

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

   if 7.2000000000000001e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e+135)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 7.2e-75)
     (/ (* 0.3333333333333333 (- (sqrt (fma (* -3.0 c) a (* b b))) b)) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+135) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 7.2e-75) {
		tmp = (0.3333333333333333 * (sqrt(fma((-3.0 * c), a, (b * b))) - b)) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e+135)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 7.2e-75)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b)) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e+135], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-75], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000001e135

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 35.6

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 35.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.6

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

   6. Applied rewrites 35.6%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if -2.1000000000000001e135 < b < 7.2000000000000001e-75

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

   if 7.2000000000000001e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e+135)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 7.2e-75)
     (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+135) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 7.2e-75) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e+135)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 7.2e-75)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e+135], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.2e-75], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000001e135

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 35.6

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 35.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 35.6

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

   6. Applied rewrites 35.6%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if -2.1000000000000001e135 < b < 7.2000000000000001e-75

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 84.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 84.9

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

   4. Applied rewrites 84.9%

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

   if 7.2000000000000001e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, -0.5, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-32)
   (* (fma (/ c (* b b)) -0.5 (/ 0.6666666666666666 a)) (- b))
   (if (<= b 5.6e-75)
     (/ (* (- (sqrt (* (* -3.0 c) a)) b) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = fma((c / (b * b)), -0.5, (0.6666666666666666 / a)) * -b;
	} else if (b <= 5.6e-75) {
		tmp = ((sqrt(((-3.0 * c) * a)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-32)
		tmp = Float64(fma(Float64(c / Float64(b * b)), -0.5, Float64(0.6666666666666666 / a)) * Float64(-b));
	elseif (b <= 5.6e-75)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(-3.0 * c) * a)) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-32], N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision], If[LessEqual[b, 5.6e-75], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999996e-32

   1. Initial program 60.2%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if -1.79999999999999996e-32 < b < 5.59999999999999996e-75

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 73.4%

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

   11. Applied rewrites 73.6%

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

   if 5.59999999999999996e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, -0.5, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-32)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 5.6e-75)
     (/ (* (- (sqrt (* (* -3.0 c) a)) b) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = ((sqrt(((-3.0 * c) * a)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.8d-32)) then
        tmp = ((-b - b) / 3.0d0) / a
    else if (b <= 5.6d-75) then
        tmp = ((sqrt((((-3.0d0) * c) * a)) - b) * 0.3333333333333333d0) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = ((Math.sqrt(((-3.0 * c) * a)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-32:
		tmp = ((-b - b) / 3.0) / a
	elif b <= 5.6e-75:
		tmp = ((math.sqrt(((-3.0 * c) * a)) - b) * 0.3333333333333333) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-32)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 5.6e-75)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(-3.0 * c) * a)) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-32)
		tmp = ((-b - b) / 3.0) / a;
	elseif (b <= 5.6e-75)
		tmp = ((sqrt(((-3.0 * c) * a)) - b) * 0.3333333333333333) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-32], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 5.6e-75], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999996e-32

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 60.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 60.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   9. Applied rewrites 90.0%

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

   if -1.79999999999999996e-32 < b < 5.59999999999999996e-75

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 73.4%

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

   11. Applied rewrites 73.6%

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

   if 5.59999999999999996e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-32)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 5.6e-75)
     (/ (* (- (sqrt (* (* c a) -3.0)) b) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = ((sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.8d-32)) then
        tmp = ((-b - b) / 3.0d0) / a
    else if (b <= 5.6d-75) then
        tmp = ((sqrt(((c * a) * (-3.0d0))) - b) * 0.3333333333333333d0) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = ((Math.sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-32:
		tmp = ((-b - b) / 3.0) / a
	elif b <= 5.6e-75:
		tmp = ((math.sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-32)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 5.6e-75)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-32)
		tmp = ((-b - b) / 3.0) / a;
	elseif (b <= 5.6e-75)
		tmp = ((sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-32], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 5.6e-75], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999996e-32

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 60.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 60.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   9. Applied rewrites 90.0%

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

   if -1.79999999999999996e-32 < b < 5.59999999999999996e-75

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if 5.59999999999999996e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-32)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 5.6e-75)
     (* (/ (- (sqrt (* (* c a) -3.0)) b) a) 0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = ((sqrt(((c * a) * -3.0)) - b) / a) * 0.3333333333333333;
	} 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 <= (-1.8d-32)) then
        tmp = ((-b - b) / 3.0d0) / a
    else if (b <= 5.6d-75) then
        tmp = ((sqrt(((c * a) * (-3.0d0))) - b) / a) * 0.3333333333333333d0
    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 <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = ((Math.sqrt(((c * a) * -3.0)) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-32:
		tmp = ((-b - b) / 3.0) / a
	elif b <= 5.6e-75:
		tmp = ((math.sqrt(((c * a) * -3.0)) - b) / a) * 0.3333333333333333
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-32)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 5.6e-75)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) / a) * 0.3333333333333333);
	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 <= -1.8e-32)
		tmp = ((-b - b) / 3.0) / a;
	elseif (b <= 5.6e-75)
		tmp = ((sqrt(((c * a) * -3.0)) - b) / a) * 0.3333333333333333;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-32], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 5.6e-75], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999996e-32

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 60.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 60.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   9. Applied rewrites 90.0%

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

   if -1.79999999999999996e-32 < b < 5.59999999999999996e-75

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   9. Applied rewrites 73.4%

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

   if 5.59999999999999996e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-32)
   (/ (/ (- (- b) b) 3.0) a)
   (if (<= b 5.6e-75)
     (* (- (sqrt (* (* c a) -3.0)) b) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = (sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.8d-32)) then
        tmp = ((-b - b) / 3.0d0) / a
    else if (b <= 5.6d-75) then
        tmp = (sqrt(((c * a) * (-3.0d0))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-32) {
		tmp = ((-b - b) / 3.0) / a;
	} else if (b <= 5.6e-75) {
		tmp = (Math.sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-32:
		tmp = ((-b - b) / 3.0) / a
	elif b <= 5.6e-75:
		tmp = (math.sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-32)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	elseif (b <= 5.6e-75)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-32)
		tmp = ((-b - b) / 3.0) / a;
	elseif (b <= 5.6e-75)
		tmp = (sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-32], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 5.6e-75], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999996e-32

   1. Initial program 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 60.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 60.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   9. Applied rewrites 90.0%

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

   if -1.79999999999999996e-32 < b < 5.59999999999999996e-75

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/r* N/A

         \[\leadsto \left(\sqrt{\mathsf{Rewrite=>}\left(lower-*.f64, \left(c \cdot a\right)\right) \cdot -3} - b\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}} \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\mathsf{Rewrite=>}\left(lower-*.f64, \left(c \cdot a\right)\right) \cdot -3} - b\right) \cdot \frac{\color{blue}{\frac{1}{3}}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{Rewrite=>}\left(lower-*.f64, \left(c \cdot a\right)\right) \cdot -3} - b\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}} \]

   9. Applied rewrites 73.4%

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

   if 5.59999999999999996e-75 < b

   1. Initial program 21.2%

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

   3. Taylor expanded in 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 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (/ (- (- b) b) 3.0) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((-b - b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-b - b) / 3.0d0) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((-b - b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = ((-b - b) / 3.0) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = ((-b - b) / 3.0) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(N[((-b) - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 66.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 66.3

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 66.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 66.3

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

   6. Applied rewrites 66.3%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.0

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

   9. Applied rewrites 73.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 38.6%

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

   3. Taylor expanded in a around 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 57.2

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

   5. Applied rewrites 57.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (* -0.6666666666666666 b) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 72.9

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

   7. Applied rewrites 72.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 38.6%

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

   3. Taylor expanded in a around 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 57.2

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

   5. Applied rewrites 57.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (* -1.5 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = b / ((-1.5d0) * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b / (-1.5 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(-1.5 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b / (-1.5 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b / N[(-1.5 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 72.8%

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

   9. Applied rewrites 72.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 38.6%

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

   3. Taylor expanded in a around 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 57.2

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

   5. Applied rewrites 57.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* (/ b a) -0.6666666666666666) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		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 <= -2e-310)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], 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 < -1.999999999999994e-310

   1. Initial program 66.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 38.6%

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

   3. Taylor expanded in a around 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 57.2

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

   5. Applied rewrites 57.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.7% 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 51.6%

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

3. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 36.0

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

5. Applied rewrites 36.0%

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

6. Final simplification 36.0%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))