Cubic critical

Percentage Accurate: 51.9% → 84.5%

Time: 16.8s

Alternatives: 14

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+135}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+135)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.05e-152)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+135) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.05e-152) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d+135)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.05d-152) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+135) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.05e-152) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e+135:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.05e-152:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+135)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.05e-152)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e+135)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.05e-152)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+135], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-152], 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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3e135

   1. Initial program 53.9%

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

   3. Taylor expanded in b around -inf 98.1%

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

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if -3e135 < b < 1.04999999999999999e-152

   1. Initial program 86.5%

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

   if 1.04999999999999999e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+135}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -650000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}}{3} - \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -650000.0)
   (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b)))
   (if (<= b 1.05e-152)
     (- (/ (/ (sqrt (* c (* a -3.0))) a) 3.0) (/ b (* 3.0 a)))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -650000.0) {
		tmp = fma(-0.6666666666666666, (b / a), (0.5 * (c / b)));
	} else if (b <= 1.05e-152) {
		tmp = ((sqrt((c * (a * -3.0))) / a) / 3.0) - (b / (3.0 * a));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -650000.0)
		tmp = fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.05e-152)
		tmp = Float64(Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) / a) / 3.0) - Float64(b / Float64(3.0 * a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -650000.0], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-152], N[(N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision] - N[(b / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e5

   1. Initial program 69.4%

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

   3. Taylor expanded in b around -inf 94.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. fma-define 94.2%

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

   5. Simplified 94.2%

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

   if -6.5e5 < b < 1.04999999999999999e-152

   1. Initial program 81.0%

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

      1. associate-*l* 80.9%

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

      2. prod-diff 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. *-commutative 80.6%

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

      6. distribute-rgt-neg-in 80.6%

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

      7. *-commutative 80.6%

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

      8. distribute-rgt-neg-in 80.6%

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

      9. metadata-eval 80.6%

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

      10. *-commutative 80.6%

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

      11. associate-*l* 80.6%

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

      12. *-commutative 80.6%

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

      13. associate-*l* 80.6%

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

   4. Applied egg-rr 80.6%

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

   5. Taylor expanded in b around 0 68.3%

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

      1. distribute-rgt-out 68.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} + -1 \cdot b}{3 \cdot a} \]

      2. *-commutative 68.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)} + -1 \cdot b}{3 \cdot a} \]

      3. metadata-eval 68.7%

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

      4. associate-*r* 68.7%

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

      5. mul-1-neg 68.7%

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

      6. unsub-neg 68.7%

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

      7. associate-*r* 68.7%

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

      8. *-commutative 68.7%

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

      9. rem-square-sqrt 0.0%

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

      10. unpow2 0.0%

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

      11. associate-*r* 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 68.7%

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

   7. Simplified 68.7%

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

      1. div-sub 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. *-commutative 68.7%

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

      5. *-commutative 68.7%

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

   9. Applied egg-rr 68.7%

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

       1. associate-/r* 68.8%

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

   11. Simplified 68.8%

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

   if 1.04999999999999999e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -650000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}}{3} - \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -10000.0)
   (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b)))
   (if (<= b 1.9e-156)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -10000.0) {
		tmp = fma(-0.6666666666666666, (b / a), (0.5 * (c / b)));
	} else if (b <= 1.9e-156) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -10000.0)
		tmp = fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.9e-156)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -10000.0], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-156], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e4

   1. Initial program 69.4%

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

   3. Taylor expanded in b around -inf 94.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. fma-define 94.2%

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

   5. Simplified 94.2%

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

   if -1e4 < b < 1.90000000000000004e-156

   1. Initial program 81.0%

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

      1. associate-*l* 80.9%

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

      2. prod-diff 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. *-commutative 80.6%

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

      6. distribute-rgt-neg-in 80.6%

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

      7. *-commutative 80.6%

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

      8. distribute-rgt-neg-in 80.6%

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

      9. metadata-eval 80.6%

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

      10. *-commutative 80.6%

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

      11. associate-*l* 80.6%

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

      12. *-commutative 80.6%

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

      13. associate-*l* 80.6%

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

   4. Applied egg-rr 80.6%

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

   5. Taylor expanded in b around 0 68.3%

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

      1. distribute-rgt-out 68.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} + -1 \cdot b}{3 \cdot a} \]

      2. *-commutative 68.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)} + -1 \cdot b}{3 \cdot a} \]

      3. metadata-eval 68.7%

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

      4. associate-*r* 68.7%

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

      5. mul-1-neg 68.7%

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

      6. unsub-neg 68.7%

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

      7. associate-*r* 68.7%

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

      8. *-commutative 68.7%

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

      9. rem-square-sqrt 0.0%

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

      10. unpow2 0.0%

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

      11. associate-*r* 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in a around 0 68.7%

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

   if 1.90000000000000004e-156 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -10000.0)
   (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b)))
   (if (<= b 1.05e-152)
     (/ (- (sqrt (* a (* c -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -10000.0) {
		tmp = fma(-0.6666666666666666, (b / a), (0.5 * (c / b)));
	} else if (b <= 1.05e-152) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -10000.0)
		tmp = fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.05e-152)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -10000.0], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-152], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e4

   1. Initial program 69.4%

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

   3. Taylor expanded in b around -inf 94.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. fma-define 94.2%

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

   5. Simplified 94.2%

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

   if -1e4 < b < 1.04999999999999999e-152

   1. Initial program 81.0%

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

      1. associate-*l* 80.9%

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

      2. prod-diff 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. *-commutative 80.6%

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

      6. distribute-rgt-neg-in 80.6%

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

      7. *-commutative 80.6%

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

      8. distribute-rgt-neg-in 80.6%

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

      9. metadata-eval 80.6%

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

      10. *-commutative 80.6%

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

      11. associate-*l* 80.6%

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

      12. *-commutative 80.6%

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

      13. associate-*l* 80.6%

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

   4. Applied egg-rr 80.6%

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

   5. Taylor expanded in b around 0 68.3%

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

      1. distribute-rgt-out 68.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} + -1 \cdot b}{3 \cdot a} \]

      2. *-commutative 68.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)} + -1 \cdot b}{3 \cdot a} \]

      3. metadata-eval 68.7%

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

      4. associate-*r* 68.7%

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

      5. mul-1-neg 68.7%

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

      6. unsub-neg 68.7%

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

      7. associate-*r* 68.7%

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

      8. *-commutative 68.7%

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

      9. rem-square-sqrt 0.0%

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

      10. unpow2 0.0%

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

      11. associate-*r* 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 68.7%

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

   7. Simplified 68.7%

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

   if 1.04999999999999999e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -19000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 10^{-152}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -19000.0)
   (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b)))
   (if (<= b 1e-152)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -19000.0) {
		tmp = fma(-0.6666666666666666, (b / a), (0.5 * (c / b)));
	} else if (b <= 1e-152) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -19000.0)
		tmp = fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b)));
	elseif (b <= 1e-152)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -19000.0], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-152], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -19000

   1. Initial program 69.4%

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

   3. Taylor expanded in b around -inf 94.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. fma-define 94.2%

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

   5. Simplified 94.2%

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

   if -19000 < b < 1.00000000000000007e-152

   1. Initial program 81.0%

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

   3. Taylor expanded in b around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

      3. *-commutative 68.7%

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

   5. Simplified 68.7%

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

   if 1.00000000000000007e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -19000:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 10^{-152}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-153}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-67)
   (+ (* 0.5 (/ c b)) (* -0.6666666666666666 (/ b a)))
   (if (<= b 9.4e-153)
     (* 0.3333333333333333 (/ (sqrt (* a (* c -3.0))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-67) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else if (b <= 9.4e-153) {
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d-67)) then
        tmp = (0.5d0 * (c / b)) + ((-0.6666666666666666d0) * (b / a))
    else if (b <= 9.4d-153) then
        tmp = 0.3333333333333333d0 * (sqrt((a * (c * (-3.0d0)))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-67) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else if (b <= 9.4e-153) {
		tmp = 0.3333333333333333 * (Math.sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-67:
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a))
	elif b <= 9.4e-153:
		tmp = 0.3333333333333333 * (math.sqrt((a * (c * -3.0))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-67)
		tmp = Float64(Float64(0.5 * Float64(c / b)) + Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 9.4e-153)
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -3.0))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-67)
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	elseif (b <= 9.4e-153)
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-67], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.4e-153], N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000002e-67

   1. Initial program 71.6%

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

   3. Taylor expanded in b around -inf 88.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -2.1000000000000002e-67 < b < 9.3999999999999998e-153

   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. associate-*l* 79.3%

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

      2. prod-diff 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*l* 79.0%

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

      5. *-commutative 79.0%

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

      6. distribute-rgt-neg-in 79.0%

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

      7. *-commutative 79.0%

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

      8. distribute-rgt-neg-in 79.0%

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

      9. metadata-eval 79.0%

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

      10. *-commutative 79.0%

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

      11. associate-*l* 79.0%

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

      12. *-commutative 79.0%

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

      13. associate-*l* 78.9%

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

   4. Applied egg-rr 78.9%

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

   5. Taylor expanded in b around 0 69.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 69.7%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      2. distribute-rgt-out 70.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      3. *-commutative 70.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)}}{a} \]

      4. metadata-eval 70.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\left(c \cdot a\right) \cdot \color{blue}{-3}}}{a} \]

      5. associate-*r* 70.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

      6. *-lft-identity 70.1%

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

      7. associate-*r* 70.0%

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

      8. *-commutative 70.0%

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

      9. rem-square-sqrt 0.0%

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

      10. unpow2 0.0%

          \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot \color{blue}{{\left(\sqrt{-3}\right)}^{2}}}}{a} \]

      11. associate-*r* 0.0%

          \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}}}{a} \]

      12. *-commutative 0.0%

          \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{a \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}}{a} \]

      13. unpow2 0.0%

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

      14. rem-square-sqrt 70.0%

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

   7. Simplified 70.0%

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

   if 9.3999999999999998e-153 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-153}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-69)
   (+ (* 0.5 (/ c b)) (* -0.6666666666666666 (/ b a)))
   (if (<= b 1.05e-152)
     (* (/ (sqrt (* c (* a -3.0))) a) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-69) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else if (b <= 1.05e-152) {
		tmp = (sqrt((c * (a * -3.0))) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.8d-69)) then
        tmp = (0.5d0 * (c / b)) + ((-0.6666666666666666d0) * (b / a))
    else if (b <= 1.05d-152) then
        tmp = (sqrt((c * (a * (-3.0d0)))) / a) * 0.3333333333333333d0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-69) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else if (b <= 1.05e-152) {
		tmp = (Math.sqrt((c * (a * -3.0))) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e-69:
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a))
	elif b <= 1.05e-152:
		tmp = (math.sqrt((c * (a * -3.0))) / a) * 0.3333333333333333
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-69)
		tmp = Float64(Float64(0.5 * Float64(c / b)) + Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.05e-152)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e-69)
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	elseif (b <= 1.05e-152)
		tmp = (sqrt((c * (a * -3.0))) / a) * 0.3333333333333333;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-69], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-152], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.79999999999999961e-69

   1. Initial program 71.6%

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

   3. Taylor expanded in b around -inf 88.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -7.79999999999999961e-69 < b < 1.04999999999999999e-152

   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. associate-*l* 79.3%

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

      2. prod-diff 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*l* 79.0%

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

      5. *-commutative 79.0%

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

      6. distribute-rgt-neg-in 79.0%

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

      7. *-commutative 79.0%

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

      8. distribute-rgt-neg-in 79.0%

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

      9. metadata-eval 79.0%

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

      10. *-commutative 79.0%

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

      11. associate-*l* 79.0%

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

      12. *-commutative 79.0%

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

      13. associate-*l* 78.9%

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

   4. Applied egg-rr 78.9%

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

   5. Taylor expanded in b around 0 69.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 69.7%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      2. distribute-rgt-out 70.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      3. *-commutative 70.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)}}{a} \]

      4. metadata-eval 70.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\left(c \cdot a\right) \cdot \color{blue}{-3}}}{a} \]

      5. associate-*r* 70.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{1 \cdot \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

      6. *-lft-identity 70.1%

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

   7. Simplified 70.1%

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

   if 1.04999999999999999e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-67)
   (+ (* 0.5 (/ c b)) (* -0.6666666666666666 (/ b a)))
   (if (<= b 1.05e-152)
     (/ (sqrt (* a (* c -3.0))) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-67) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else if (b <= 1.05e-152) {
		tmp = sqrt((a * (c * -3.0))) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6d-67)) then
        tmp = (0.5d0 * (c / b)) + ((-0.6666666666666666d0) * (b / a))
    else if (b <= 1.05d-152) then
        tmp = sqrt((a * (c * (-3.0d0)))) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-67) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else if (b <= 1.05e-152) {
		tmp = Math.sqrt((a * (c * -3.0))) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-67:
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a))
	elif b <= 1.05e-152:
		tmp = math.sqrt((a * (c * -3.0))) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-67)
		tmp = Float64(Float64(0.5 * Float64(c / b)) + Float64(-0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.05e-152)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-67)
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	elseif (b <= 1.05e-152)
		tmp = sqrt((a * (c * -3.0))) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-67], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-152], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000065e-67

   1. Initial program 71.6%

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

   3. Taylor expanded in b around -inf 88.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -6.00000000000000065e-67 < b < 1.04999999999999999e-152

   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. prod-diff 79.0%

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

      2. distribute-rgt-neg-in 79.0%

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

      3. *-commutative 79.0%

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

      4. distribute-rgt-neg-in 79.0%

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

      5. metadata-eval 79.0%

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

      6. *-commutative 79.0%

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

      7. fma-undefine 79.0%

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

      8. distribute-lft-neg-in 79.0%

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

      9. distribute-rgt-neg-in 79.0%

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

      10. *-commutative 79.0%

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

      11. distribute-rgt-neg-in 79.0%

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

      12. metadata-eval 79.0%

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

      13. *-commutative 79.0%

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

      14. associate-*l* 78.9%

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

   4. Applied egg-rr 78.9%

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

   5. Taylor expanded in b around 0 69.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval 69.7%

         \[\leadsto \color{blue}{\frac{1}{3}} \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right) \]

      2. associate-*l/ 69.7%

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      3. distribute-rgt-out 70.0%

         \[\leadsto \frac{1}{3} \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      4. *-commutative 70.0%

         \[\leadsto \frac{1}{3} \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)}}{a} \]

      5. metadata-eval 70.0%

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

      6. associate-*r* 70.1%

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

      7. *-lft-identity 70.1%

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

      8. times-frac 70.1%

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

      9. *-lft-identity 70.1%

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

      10. *-commutative 70.1%

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

      11. associate-*r* 70.1%

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

      12. *-commutative 70.1%

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

      13. associate-*r* 70.1%

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

   7. Simplified 70.1%

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

   if 1.04999999999999999e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-66)
   (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b)))
   (if (<= b 1.05e-152)
     (/ (sqrt (* a (* c -3.0))) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e-66) {
		tmp = fma(-0.6666666666666666, (b / a), (0.5 * (c / b)));
	} else if (b <= 1.05e-152) {
		tmp = sqrt((a * (c * -3.0))) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.12e-66)
		tmp = fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.05e-152)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e-66], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-152], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.12000000000000004e-66

   1. Initial program 71.6%

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

   3. Taylor expanded in b around -inf 88.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. fma-define 88.8%

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

   5. Simplified 88.8%

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

   if -1.12000000000000004e-66 < b < 1.04999999999999999e-152

   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. prod-diff 79.0%

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

      2. distribute-rgt-neg-in 79.0%

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

      3. *-commutative 79.0%

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

      4. distribute-rgt-neg-in 79.0%

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

      5. metadata-eval 79.0%

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

      6. *-commutative 79.0%

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

      7. fma-undefine 79.0%

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

      8. distribute-lft-neg-in 79.0%

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

      9. distribute-rgt-neg-in 79.0%

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

      10. *-commutative 79.0%

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

      11. distribute-rgt-neg-in 79.0%

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

      12. metadata-eval 79.0%

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

      13. *-commutative 79.0%

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

      14. associate-*l* 78.9%

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

   4. Applied egg-rr 78.9%

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

   5. Taylor expanded in b around 0 69.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval 69.7%

         \[\leadsto \color{blue}{\frac{1}{3}} \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right) \]

      2. associate-*l/ 69.7%

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      3. distribute-rgt-out 70.0%

         \[\leadsto \frac{1}{3} \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      4. *-commutative 70.0%

         \[\leadsto \frac{1}{3} \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-6 + 3\right)}}{a} \]

      5. metadata-eval 70.0%

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

      6. associate-*r* 70.1%

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

      7. *-lft-identity 70.1%

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

      8. times-frac 70.1%

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

      9. *-lft-identity 70.1%

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

      10. *-commutative 70.1%

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

      11. associate-*r* 70.1%

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

      12. *-commutative 70.1%

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

      13. associate-*r* 70.1%

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

   7. Simplified 70.1%

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

   if 1.04999999999999999e-152 < b

   1. Initial program 22.9%

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

   3. Taylor expanded in b around inf 64.8%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in a around 0 82.7%

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

      1. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-152}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (0.5d0 * (c / b)) + ((-0.6666666666666666d0) * (b / a))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf 71.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 31.1%

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

   3. Taylor expanded in b around inf 55.4%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 60.9%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in a around 0 71.3%

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

      1. associate-*r/ 71.3%

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

   8. Simplified 71.3%

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

4. Final simplification 71.2%

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

Alternative 11: 67.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf 70.7%

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

      1. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.1%

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

   3. Taylor expanded in b around inf 55.4%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 60.9%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in a around 0 71.3%

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

      1. associate-*r/ 71.3%

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

   8. Simplified 71.3%

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

4. Final simplification 71.0%

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

Alternative 12: 67.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf 70.7%

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

      1. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.1%

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

   3. Taylor expanded in b around inf 71.3%

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

4. Final simplification 71.0%

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

Alternative 13: 67.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf 70.7%

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

      1. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.1%

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

   3. Taylor expanded in b around inf 55.4%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-/l* 60.9%

         \[\leadsto \frac{-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in a around 0 71.3%

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

      1. associate-*r/ 71.3%

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

   8. Simplified 71.3%

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

4. Final simplification 71.0%

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

Alternative 14: 35.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in b around inf 36.7%

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

4. Final simplification 36.7%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))