Cubic critical

Percentage Accurate: 52.5% → 85.7%

Time: 38.5s

Alternatives: 12

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: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e+92)
   (/ b (* a -1.5))
   (if (<= b 1.55e-59)
     (/ (/ (- b (sqrt (fma -3.0 (* a c) (pow b 2.0)))) a) -3.0)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e+92) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.55e-59) {
		tmp = ((b - sqrt(fma(-3.0, (a * c), pow(b, 2.0)))) / a) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e+92)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.55e-59)
		tmp = Float64(Float64(Float64(b - sqrt(fma(-3.0, Float64(a * c), (b ^ 2.0)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e+92], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-59], N[(N[(N[(b - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2999999999999999e92

   1. Initial program 53.2%

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

      1. sqr-neg 53.2%

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

      2. sqr-neg 53.2%

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

      3. associate-*l* 53.2%

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

   3. Simplified 53.2%

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

      1. frac-2neg 53.2%

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

      2. div-inv 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. fma-undefine 53.1%

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

      2. unpow2 53.1%

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

      3. associate-*l* 53.1%

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

      4. *-commutative 53.1%

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

      5. +-commutative 53.1%

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

      6. fma-define 53.1%

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

      7. associate-/r* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in b around -inf 95.4%

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

       1. *-commutative 95.4%

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

       2. associate-*l/ 95.3%

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

       3. associate-/l* 95.3%

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

   11. Simplified 95.3%

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

       1. clear-num 95.3%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 95.5%

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

       3. div-inv 95.6%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 95.6%

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

   13. Applied egg-rr 95.6%

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

   if -1.2999999999999999e92 < b < 1.55e-59

   1. Initial program 80.2%

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

      1. sqr-neg 80.2%

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

      2. sqr-neg 80.2%

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

      3. associate-*l* 80.1%

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

   3. Simplified 80.1%

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

      1. frac-2neg 80.1%

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

      2. div-inv 80.1%

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

   6. Applied egg-rr 80.0%

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

      1. fma-undefine 80.0%

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

      2. unpow2 80.0%

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

      3. associate-*l* 80.1%

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

      4. *-commutative 80.1%

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

      5. +-commutative 80.1%

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

      6. fma-define 80.1%

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

      7. associate-/r* 80.1%

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

   8. Simplified 80.1%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.1%

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

   10. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   11. Step-by-step derivation

       1. associate-*r/ 80.3%

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

       2. *-rgt-identity 80.3%

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

       3. *-commutative 80.3%

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

   12. Simplified 80.3%

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

   if 1.55e-59 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 92.4%

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

      1. *-commutative 92.4%

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

   7. Simplified 92.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{b - \sqrt{{b}^{2} + c \cdot \left(a \cdot -3\right)}}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e+92)
   (/ b (* a -1.5))
   (if (<= b 9.5e-60)
     (/ (/ 1.0 (/ a (- b (sqrt (+ (pow b 2.0) (* c (* a -3.0))))))) -3.0)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e+92) {
		tmp = b / (a * -1.5);
	} else if (b <= 9.5e-60) {
		tmp = (1.0 / (a / (b - sqrt((pow(b, 2.0) + (c * (a * -3.0))))))) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d+92)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 9.5d-60) then
        tmp = (1.0d0 / (a / (b - sqrt(((b ** 2.0d0) + (c * (a * (-3.0d0)))))))) / (-3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e+92) {
		tmp = b / (a * -1.5);
	} else if (b <= 9.5e-60) {
		tmp = (1.0 / (a / (b - Math.sqrt((Math.pow(b, 2.0) + (c * (a * -3.0))))))) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e+92:
		tmp = b / (a * -1.5)
	elif b <= 9.5e-60:
		tmp = (1.0 / (a / (b - math.sqrt((math.pow(b, 2.0) + (c * (a * -3.0))))))) / -3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e+92)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 9.5e-60)
		tmp = Float64(Float64(1.0 / Float64(a / Float64(b - sqrt(Float64((b ^ 2.0) + Float64(c * Float64(a * -3.0))))))) / -3.0);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e+92)
		tmp = b / (a * -1.5);
	elseif (b <= 9.5e-60)
		tmp = (1.0 / (a / (b - sqrt(((b ^ 2.0) + (c * (a * -3.0))))))) / -3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e+92], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-60], N[(N[(1.0 / N[(a / N[(b - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999999e92

   1. Initial program 53.2%

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

      1. sqr-neg 53.2%

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

      2. sqr-neg 53.2%

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

      3. associate-*l* 53.2%

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

   3. Simplified 53.2%

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

      1. frac-2neg 53.2%

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

      2. div-inv 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. fma-undefine 53.1%

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

      2. unpow2 53.1%

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

      3. associate-*l* 53.1%

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

      4. *-commutative 53.1%

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

      5. +-commutative 53.1%

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

      6. fma-define 53.1%

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

      7. associate-/r* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in b around -inf 95.4%

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

       1. *-commutative 95.4%

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

       2. associate-*l/ 95.3%

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

       3. associate-/l* 95.3%

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

   11. Simplified 95.3%

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

       1. clear-num 95.3%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 95.5%

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

       3. div-inv 95.6%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 95.6%

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

   13. Applied egg-rr 95.6%

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

   if -1.14999999999999999e92 < b < 9.49999999999999958e-60

   1. Initial program 80.2%

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

      1. sqr-neg 80.2%

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

      2. sqr-neg 80.2%

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

      3. associate-*l* 80.1%

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

   3. Simplified 80.1%

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

      1. frac-2neg 80.1%

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

      2. div-inv 80.1%

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

   6. Applied egg-rr 80.0%

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

      1. fma-undefine 80.0%

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

      2. unpow2 80.0%

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

      3. associate-*l* 80.1%

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

      4. *-commutative 80.1%

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

      5. +-commutative 80.1%

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

      6. fma-define 80.1%

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

      7. associate-/r* 80.1%

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

   8. Simplified 80.1%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.1%

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

   10. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   11. Step-by-step derivation

       1. associate-*r/ 80.3%

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

       2. *-rgt-identity 80.3%

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

       3. *-commutative 80.3%

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

   12. Simplified 80.3%

       \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}{a}}{-3}} \]
   13. Step-by-step derivation

       1. clear-num 80.2%

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

       2. inv-pow 80.2%

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

   14. Applied egg-rr 80.2%

       \[\leadsto \frac{\color{blue}{{\left(\frac{a}{b - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}\right)}^{-1}}}{-3} \]
   15. Step-by-step derivation

       1. unpow-1 80.2%

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

   16. Simplified 80.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{a}{b - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}}}}{-3} \]
   17. Step-by-step derivation

       1. fma-undefine 80.2%

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

       2. +-commutative 80.2%

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

       3. *-commutative 80.2%

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

       4. associate-*r* 80.2%

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

   18. Applied egg-rr 80.2%

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

   if 9.49999999999999958e-60 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 92.4%

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

      1. *-commutative 92.4%

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

   7. Simplified 92.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{b - \sqrt{{b}^{2} + c \cdot \left(a \cdot -3\right)}}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+98)
   (/ b (* a -1.5))
   (if (<= b 2.15e-61)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+98) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.15e-61) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d+98)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.15d-61) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+98) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.15e-61) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e+98:
		tmp = b / (a * -1.5)
	elif b <= 2.15e-61:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+98)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.15e-61)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e+98)
		tmp = b / (a * -1.5);
	elseif (b <= 2.15e-61)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+98], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.15e-61], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e98

   1. Initial program 50.1%

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

      1. sqr-neg 50.1%

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

      2. sqr-neg 50.1%

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

      3. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. frac-2neg 50.1%

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

      2. div-inv 50.1%

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

   6. Applied egg-rr 50.1%

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

      1. fma-undefine 50.1%

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

      2. unpow2 50.1%

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

      3. associate-*l* 50.1%

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

      4. *-commutative 50.1%

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

      5. +-commutative 50.1%

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

      6. fma-define 50.1%

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

      7. associate-/r* 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in b around -inf 95.1%

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

       1. *-commutative 95.1%

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

       2. associate-*l/ 95.1%

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

       3. associate-/l* 95.0%

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

   11. Simplified 95.0%

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

       1. clear-num 95.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 95.2%

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

       3. div-inv 95.3%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 95.3%

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

   13. Applied egg-rr 95.3%

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

   if -3.0000000000000001e98 < b < 2.1500000000000002e-61

   1. Initial program 80.9%

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

   if 2.1500000000000002e-61 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 92.4%

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

      1. *-commutative 92.4%

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

   7. Simplified 92.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e+98)
   (/ b (* a -1.5))
   (if (<= b 1.4e-62)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e+98) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.4e-62) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d+98)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.4d-62) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e+98) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.4e-62) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e+98:
		tmp = b / (a * -1.5)
	elif b <= 1.4e-62:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e+98)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.4e-62)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e+98)
		tmp = b / (a * -1.5);
	elseif (b <= 1.4e-62)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e+98], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-62], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000001e98

   1. Initial program 50.1%

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

      1. sqr-neg 50.1%

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

      2. sqr-neg 50.1%

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

      3. associate-*l* 50.1%

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

   3. Simplified 50.1%

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

      1. frac-2neg 50.1%

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

      2. div-inv 50.1%

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

   6. Applied egg-rr 50.1%

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

      1. fma-undefine 50.1%

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

      2. unpow2 50.1%

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

      3. associate-*l* 50.1%

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

      4. *-commutative 50.1%

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

      5. +-commutative 50.1%

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

      6. fma-define 50.1%

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

      7. associate-/r* 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in b around -inf 95.1%

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

       1. *-commutative 95.1%

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

       2. associate-*l/ 95.1%

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

       3. associate-/l* 95.0%

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

   11. Simplified 95.0%

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

       1. clear-num 95.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 95.2%

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

       3. div-inv 95.3%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 95.3%

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

   13. Applied egg-rr 95.3%

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

   if -8.2000000000000001e98 < b < 1.40000000000000001e-62

   1. Initial program 80.9%

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

      1. sqr-neg 80.9%

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

      2. sqr-neg 80.9%

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

      3. associate-*l* 80.8%

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

   3. Simplified 80.8%

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

   if 1.40000000000000001e-62 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 92.4%

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

      1. *-commutative 92.4%

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

   7. Simplified 92.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-36}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e-36)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 7.8e-59)
     (/ (/ (- b (sqrt (* -3.0 (* a c)))) a) -3.0)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-36) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 7.8e-59) {
		tmp = ((b - sqrt((-3.0 * (a * c)))) / a) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.3d-36)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 7.8d-59) then
        tmp = ((b - sqrt(((-3.0d0) * (a * c)))) / a) / (-3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-36) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 7.8e-59) {
		tmp = ((b - Math.sqrt((-3.0 * (a * c)))) / a) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e-36:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 7.8e-59:
		tmp = ((b - math.sqrt((-3.0 * (a * c)))) / a) / -3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e-36)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 7.8e-59)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(-3.0 * Float64(a * c)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.3e-36)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 7.8e-59)
		tmp = ((b - sqrt((-3.0 * (a * c)))) / a) / -3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e-36], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.8e-59], N[(N[(N[(b - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3e-36

   1. Initial program 65.4%

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

      1. sqr-neg 65.4%

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

      2. sqr-neg 65.4%

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

      3. associate-*l* 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in b around -inf 91.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 91.3%

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

   if -1.3e-36 < b < 7.80000000000000038e-59

   1. Initial program 75.7%

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

      1. sqr-neg 75.7%

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

      2. sqr-neg 75.7%

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

      3. associate-*l* 75.7%

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

   3. Simplified 75.7%

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

      1. frac-2neg 75.7%

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

      2. div-inv 75.6%

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

   6. Applied egg-rr 75.5%

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

      1. fma-undefine 75.5%

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

      2. unpow2 75.5%

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

      3. associate-*l* 75.6%

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

      4. *-commutative 75.6%

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

      5. +-commutative 75.6%

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

      6. fma-define 75.6%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 75.7%

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

   10. Applied egg-rr 75.7%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   11. Step-by-step derivation

       1. associate-*r/ 75.8%

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

       2. *-rgt-identity 75.8%

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

       3. *-commutative 75.8%

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

   12. Simplified 75.8%

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

   13. Taylor expanded in c around inf 66.7%

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

   if 7.80000000000000038e-59 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 92.4%

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

      1. *-commutative 92.4%

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

   7. Simplified 92.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-36}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 69.6%

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

      1. sqr-neg 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 68.5%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 70.2%

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

   if -9.999999999999969e-311 < b

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. associate-*l* 31.7%

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

   3. Simplified 31.7%

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

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ b (* a -1.5)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = b / (a * -1.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 69.6%

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

      1. sqr-neg 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

      1. frac-2neg 69.6%

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

      2. div-inv 69.6%

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

   6. Applied egg-rr 69.5%

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

      1. fma-undefine 69.5%

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

      2. unpow2 69.5%

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

      3. associate-*l* 69.6%

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

      4. *-commutative 69.6%

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

      5. +-commutative 69.6%

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

      6. fma-define 69.6%

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

      7. associate-/r* 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in b around -inf 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-*l/ 69.5%

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

       3. associate-/l* 69.5%

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

   11. Simplified 69.5%

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

       1. clear-num 69.4%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 69.6%

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

       3. div-inv 69.6%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 69.6%

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

   13. Applied egg-rr 69.6%

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

   if -9.999999999999969e-311 < b

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. associate-*l* 31.7%

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

   3. Simplified 31.7%

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

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

Alternative 8: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 69.6%

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

      1. sqr-neg 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   if -9.999999999999969e-311 < b

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. associate-*l* 31.7%

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

   3. Simplified 31.7%

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

   5. Taylor expanded in b around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1e-309) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		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 <= 1d-309) 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 <= 1e-309) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-309:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-309)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1e-309)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-309], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.000000000000002e-309

   1. Initial program 69.6%

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

      1. sqr-neg 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   if 1.000000000000002e-309 < b

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. associate-*l* 31.7%

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

   3. Simplified 31.7%

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

      1. frac-2neg 31.7%

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

      2. div-inv 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. fma-undefine 31.7%

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

      2. unpow2 31.7%

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

      3. associate-*l* 31.7%

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

      4. *-commutative 31.7%

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

      5. +-commutative 31.7%

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

      6. fma-define 31.7%

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

      7. associate-/r* 31.7%

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

   8. Simplified 31.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 31.7%

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

   10. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   11. Step-by-step derivation

       1. associate-*r/ 31.7%

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

       2. *-rgt-identity 31.7%

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

       3. *-commutative 31.7%

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

   12. Simplified 31.7%

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

   13. Taylor expanded in b around inf 68.7%

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

       1. associate-*r/ 68.7%

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

       2. *-commutative 68.7%

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

       3. associate-/l* 68.5%

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

   15. Simplified 68.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1e-309) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1d-309) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-309:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-309)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1e-309)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-309], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.000000000000002e-309

   1. Initial program 69.6%

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

      1. sqr-neg 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

      1. frac-2neg 69.6%

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

      2. div-inv 69.6%

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

   6. Applied egg-rr 69.5%

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

      1. fma-undefine 69.5%

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

      2. unpow2 69.5%

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

      3. associate-*l* 69.6%

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

      4. *-commutative 69.6%

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

      5. +-commutative 69.6%

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

      6. fma-define 69.6%

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

      7. associate-/r* 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in b around -inf 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-*l/ 69.5%

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

       3. associate-/l* 69.5%

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

   11. Simplified 69.5%

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

   if 1.000000000000002e-309 < b

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. associate-*l* 31.7%

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

   3. Simplified 31.7%

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

      1. frac-2neg 31.7%

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

      2. div-inv 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. fma-undefine 31.7%

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

      2. unpow2 31.7%

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

      3. associate-*l* 31.7%

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

      4. *-commutative 31.7%

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

      5. +-commutative 31.7%

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

      6. fma-define 31.7%

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

      7. associate-/r* 31.7%

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

   8. Simplified 31.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{\frac{1}{a}}{-3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 31.7%

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

   10. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)}\right) \cdot \frac{1}{a}}{-3}} \]
   11. Step-by-step derivation

       1. associate-*r/ 31.7%

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

       2. *-rgt-identity 31.7%

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

       3. *-commutative 31.7%

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

   12. Simplified 31.7%

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

   13. Taylor expanded in b around inf 68.7%

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

       1. associate-*r/ 68.7%

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

       2. *-commutative 68.7%

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

       3. associate-/l* 68.5%

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

   15. Simplified 68.5%

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

Alternative 11: 35.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. sqr-neg 52.5%

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

   2. sqr-neg 52.5%

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

   3. associate-*l* 52.4%

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

3. Simplified 52.4%

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

   1. frac-2neg 52.4%

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

   2. div-inv 52.4%

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

6. Applied egg-rr 52.4%

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

   1. fma-undefine 52.4%

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

   2. unpow2 52.4%

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

   3. associate-*l* 52.4%

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

   4. *-commutative 52.4%

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

   5. +-commutative 52.4%

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

   6. fma-define 52.4%

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

   7. associate-/r* 52.4%

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

8. Simplified 52.4%

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

9. Taylor expanded in b around -inf 39.2%

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

    1. *-commutative 39.2%

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

    2. associate-*l/ 39.2%

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

    3. associate-/l* 39.2%

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

11. Simplified 39.2%

    \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
12. Add Preprocessing

Alternative 12: 2.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (b / a) * 0.6666666666666666d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. sqr-neg 52.5%

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

   2. sqr-neg 52.5%

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

   3. associate-*l* 52.4%

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

3. Simplified 52.4%

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

   1. *-un-lft-identity 52.4%

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

   2. times-frac 52.5%

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

   3. metadata-eval 52.5%

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

   4. add-sqr-sqrt 38.1%

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

   5. sqrt-unprod 50.7%

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

   6. sqr-neg 50.7%

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

   7. sqrt-prod 12.9%

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

   8. add-sqr-sqrt 33.3%

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

   9. fma-neg 33.3%

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

   10. distribute-lft-neg-in 33.3%

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

   11. *-commutative 33.3%

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

   12. associate-*r* 33.3%

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

   13. metadata-eval 33.3%

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

6. Applied egg-rr 33.3%

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

7. Taylor expanded in b around inf 2.4%

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

8. Final simplification 2.4%

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

Reproduce

herbie shell --seed 2024143 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))