Cubic critical

Percentage Accurate: 51.2% → 84.9%

Time: 13.7s

Alternatives: 17

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+152)
   (* (/ b 3.0) (/ -2.0 a))
   (if (<= b 9.2e-10)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+152) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 9.2e-10) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+152)) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else if (b <= 9.2d-10) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+152) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 9.2e-10) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+152:
		tmp = (b / 3.0) * (-2.0 / a)
	elif b <= 9.2e-10:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+152)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	elseif (b <= 9.2e-10)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+152)
		tmp = (b / 3.0) * (-2.0 / a);
	elseif (b <= 9.2e-10)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+152], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-10], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e152

   1. Initial program 49.6%

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

   2. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -2.0000000000000001e152 < b < 9.20000000000000028e-10

   1. Initial program 80.7%

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

   if 9.20000000000000028e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+151)
   (* (/ b 3.0) (/ -2.0 a))
   (if (<= b 9.2e-10)
     (* -0.3333333333333333 (/ (- b (sqrt (- (* b b) (* 3.0 (* a c))))) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+151) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 9.2e-10) {
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.8d+151)) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else if (b <= 9.2d-10) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((b * b) - (3.0d0 * (a * c))))) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+151) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 9.2e-10) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+151:
		tmp = (b / 3.0) * (-2.0 / a)
	elif b <= 9.2e-10:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((b * b) - (3.0 * (a * c))))) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+151)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	elseif (b <= 9.2e-10)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c))))) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+151)
		tmp = (b / 3.0) * (-2.0 / a);
	elseif (b <= 9.2e-10)
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+151], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-10], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.79999999999999987e151

   1. Initial program 49.6%

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

   2. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -2.79999999999999987e151 < b < 9.20000000000000028e-10

   1. Initial program 80.7%

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

      1. /-rgt-identity 80.7%

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

      2. metadata-eval 80.7%

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

      3. associate-/l* 80.7%

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

      4. associate-*r/ 80.6%

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

      5. *-commutative 80.6%

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

      6. associate-*l/ 80.7%

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

      7. associate-*r/ 80.7%

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

      8. metadata-eval 80.7%

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

      9. metadata-eval 80.7%

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

      10. times-frac 80.7%

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

      11. neg-mul-1 80.7%

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

      12. distribute-rgt-neg-in 80.7%

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

      13. times-frac 80.5%

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

      14. metadata-eval 80.5%

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

      15. neg-mul-1 80.5%

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

   3. Simplified 80.5%

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

      1. fma-udef 80.5%

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

      2. associate-*r* 80.5%

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

      3. *-commutative 80.5%

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

      4. metadata-eval 80.5%

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

      5. cancel-sign-sub-inv 80.5%

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

   5. Applied egg-rr 80.5%

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

   if 9.20000000000000028e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e+146)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 7.5e-10)
     (* (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (/ 0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+146) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 7.5e-10) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.8d+146)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 7.5d-10) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+146) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 7.5e-10) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e+146:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 7.5e-10:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e+146)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 7.5e-10)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e+146)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 7.5e-10)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e+146], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-10], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.8e146

   1. Initial program 50.6%

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

   2. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   if -7.8e146 < b < 7.49999999999999995e-10

   1. Initial program 80.6%

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

      1. neg-sub0 80.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 80.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 80.6%

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

      4. neg-mul-1 80.6%

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

      5. associate-*r/ 80.6%

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

      6. *-commutative 80.6%

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

      7. metadata-eval 80.6%

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

      8. metadata-eval 80.6%

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

      9. times-frac 80.6%

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

      10. *-commutative 80.6%

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

      11. times-frac 80.4%

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

   3. Simplified 80.4%

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

      1. fma-udef 80.4%

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

      2. associate-*r* 80.4%

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

      3. *-commutative 80.4%

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

      4. metadata-eval 80.4%

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

      5. cancel-sign-sub-inv 80.4%

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

   5. Applied egg-rr 80.4%

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

   if 7.49999999999999995e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e+151)
   (* (/ b 3.0) (/ -2.0 a))
   (if (<= b 7e-10)
     (/ (* (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) 0.3333333333333333) a)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e+151) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 7e-10) {
		tmp = ((sqrt(((b * b) - (3.0 * (a * c)))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.7d+151)) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else if (b <= 7d-10) then
        tmp = ((sqrt(((b * b) - (3.0d0 * (a * c)))) - b) * 0.3333333333333333d0) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e+151) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 7e-10) {
		tmp = ((Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.7e+151:
		tmp = (b / 3.0) * (-2.0 / a)
	elif b <= 7e-10:
		tmp = ((math.sqrt(((b * b) - (3.0 * (a * c)))) - b) * 0.3333333333333333) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.7e+151)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	elseif (b <= 7e-10)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.7e+151)
		tmp = (b / 3.0) * (-2.0 / a);
	elseif (b <= 7e-10)
		tmp = ((sqrt(((b * b) - (3.0 * (a * c)))) - b) * 0.3333333333333333) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.7e+151], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-10], N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7000000000000001e151

   1. Initial program 49.6%

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

   2. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -2.7000000000000001e151 < b < 6.99999999999999961e-10

   1. Initial program 80.7%

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

      1. neg-sub0 80.7%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 80.7%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 80.7%

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

      4. neg-mul-1 80.7%

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

      5. associate-*r/ 80.7%

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

      6. *-commutative 80.7%

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

      7. metadata-eval 80.7%

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

      8. metadata-eval 80.7%

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

      9. times-frac 80.7%

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

      10. *-commutative 80.7%

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

      11. times-frac 80.6%

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

   3. Simplified 80.5%

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

      1. fma-udef 80.5%

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

      2. associate-*r* 80.5%

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

      3. *-commutative 80.5%

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

      4. metadata-eval 80.5%

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

      5. cancel-sign-sub-inv 80.5%

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

   5. Applied egg-rr 80.5%

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

      1. associate-*r/ 80.6%

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

   7. Applied egg-rr 80.6%

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

   if 6.99999999999999961e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.4e+151)
   (* (/ b 3.0) (/ -2.0 a))
   (if (<= b 6e-9)
     (/ (- (sqrt (+ (* b b) (* a (* c -3.0)))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e+151) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 6e-9) {
		tmp = (sqrt(((b * b) + (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.4d+151)) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else if (b <= 6d-9) then
        tmp = (sqrt(((b * b) + (a * (c * (-3.0d0))))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e+151) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 6e-9) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.4e+151:
		tmp = (b / 3.0) * (-2.0 / a)
	elif b <= 6e-9:
		tmp = (math.sqrt(((b * b) + (a * (c * -3.0)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.4e+151)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	elseif (b <= 6e-9)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.4e+151)
		tmp = (b / 3.0) * (-2.0 / a);
	elseif (b <= 6e-9)
		tmp = (sqrt(((b * b) + (a * (c * -3.0)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.4e+151], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-9], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.39999999999999988e151

   1. Initial program 49.6%

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

   2. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -6.39999999999999988e151 < b < 5.99999999999999996e-9

   1. Initial program 80.7%

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

      1. associate-*r* 80.6%

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

      2. cancel-sign-sub-inv 80.6%

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

      3. metadata-eval 80.6%

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

      4. *-commutative 80.6%

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

      5. associate-*r* 80.6%

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

   3. Applied egg-rr 80.6%

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

   if 5.99999999999999996e-9 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+152)
   (* (/ b 3.0) (/ -2.0 a))
   (if (<= b 8e-10)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+152) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 8e-10) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d+152)) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else if (b <= 8d-10) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+152) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else if (b <= 8e-10) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+152:
		tmp = (b / 3.0) * (-2.0 / a)
	elif b <= 8e-10:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+152)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	elseif (b <= 8e-10)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+152)
		tmp = (b / 3.0) * (-2.0 / a);
	elseif (b <= 8e-10)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+152], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-10], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999993e152

   1. Initial program 49.6%

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

   2. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   if -8.4999999999999993e152 < b < 8.00000000000000029e-10

   1. Initial program 80.7%

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

      1. neg-sub0 80.7%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 80.7%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 80.7%

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

      4. neg-mul-1 80.7%

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

      5. associate-*r/ 80.7%

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

      6. metadata-eval 80.7%

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

      7. metadata-eval 80.7%

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

      8. times-frac 80.7%

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

      9. *-commutative 80.7%

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

      10. times-frac 80.6%

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

      11. associate-*l/ 80.7%

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

   3. Simplified 80.6%

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

   if 8.00000000000000029e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.62e-74)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 7.2e-10)
     (* 0.3333333333333333 (/ (- (sqrt (* c (* a -3.0))) b) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7.2e-10) {
		tmp = 0.3333333333333333 * ((sqrt((c * (a * -3.0))) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.62d-74)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 7.2d-10) then
        tmp = 0.3333333333333333d0 * ((sqrt((c * (a * (-3.0d0)))) - b) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7.2e-10) {
		tmp = 0.3333333333333333 * ((Math.sqrt((c * (a * -3.0))) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.62e-74:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 7.2e-10:
		tmp = 0.3333333333333333 * ((math.sqrt((c * (a * -3.0))) - b) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.62e-74)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 7.2e-10)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.62e-74)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 7.2e-10)
		tmp = 0.3333333333333333 * ((sqrt((c * (a * -3.0))) - b) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.62e-74], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-10], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.62000000000000007e-74

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.6%

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

   4. Applied egg-rr 90.6%

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

   if -1.62000000000000007e-74 < b < 7.2e-10

   1. Initial program 74.0%

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

   2. Taylor expanded in b around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. *-commutative 67.8%

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

      3. *-commutative 67.8%

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

      4. associate-*l* 68.0%

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

   4. Simplified 68.0%

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

      1. pow1/2 68.0%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(c \cdot \left(a \cdot -3\right)\right)}^{0.5}}}{3 \cdot a} \]

      2. *-commutative 68.0%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(\left(a \cdot -3\right) \cdot c\right)}}^{0.5}}{3 \cdot a} \]

      3. unpow-prod-down 47.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(a \cdot -3\right)}^{0.5} \cdot {c}^{0.5}}}{3 \cdot a} \]

      4. pow1/2 47.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{a \cdot -3}} \cdot {c}^{0.5}}{3 \cdot a} \]

      5. pow1/2 47.8%

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

   6. Applied egg-rr 47.8%

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

      1. clear-num 47.8%

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

      2. inv-pow 47.8%

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

      3. neg-mul-1 47.8%

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

      4. fma-def 47.8%

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

      5. sqrt-unprod 67.9%

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

   8. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.8%

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

      3. associate-/r/ 67.8%

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

      4. metadata-eval 67.8%

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

      5. fma-udef 67.8%

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

      6. *-commutative 67.8%

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

      7. fma-def 67.8%

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

      8. *-commutative 67.8%

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

   10. Simplified 67.8%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]
   11. Step-by-step derivation

       1. div-inv 67.8%

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

   12. Applied egg-rr 67.8%

       \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\mathsf{fma}\left(b, -1, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{a}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 67.8%

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

       2. *-rgt-identity 67.8%

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

       3. fma-udef 67.8%

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

       4. *-commutative 67.8%

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

       5. neg-mul-1 67.8%

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

       6. +-commutative 67.8%

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

       7. unsub-neg 67.8%

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

   14. Simplified 67.8%

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

   if 7.2e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e-74)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 7e-10)
     (* (/ 0.3333333333333333 a) (- (sqrt (* c (* a -3.0))) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7e-10) {
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.45d-74)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 7d-10) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((c * (a * (-3.0d0)))) - b)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7e-10) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((c * (a * -3.0))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e-74:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 7e-10:
		tmp = (0.3333333333333333 / a) * (math.sqrt((c * (a * -3.0))) - b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e-74)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 7e-10)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(c * Float64(a * -3.0))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.45e-74)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 7e-10)
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e-74], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-10], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45e-74

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.6%

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

   4. Applied egg-rr 90.6%

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

   if -1.45e-74 < b < 6.99999999999999961e-10

   1. Initial program 74.0%

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

   2. Taylor expanded in b around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. *-commutative 67.8%

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

      3. *-commutative 67.8%

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

      4. associate-*l* 68.0%

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

   4. Simplified 68.0%

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

      1. pow1/2 68.0%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(c \cdot \left(a \cdot -3\right)\right)}^{0.5}}}{3 \cdot a} \]

      2. *-commutative 68.0%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(\left(a \cdot -3\right) \cdot c\right)}}^{0.5}}{3 \cdot a} \]

      3. unpow-prod-down 47.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(a \cdot -3\right)}^{0.5} \cdot {c}^{0.5}}}{3 \cdot a} \]

      4. pow1/2 47.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{a \cdot -3}} \cdot {c}^{0.5}}{3 \cdot a} \]

      5. pow1/2 47.8%

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

   6. Applied egg-rr 47.8%

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

      1. clear-num 47.8%

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

      2. inv-pow 47.8%

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

      3. neg-mul-1 47.8%

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

      4. fma-def 47.8%

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

      5. sqrt-unprod 67.9%

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

   8. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.8%

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

      3. associate-/r/ 67.8%

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

      4. metadata-eval 67.8%

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

      5. fma-udef 67.8%

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

      6. *-commutative 67.8%

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

      7. fma-def 67.8%

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

      8. *-commutative 67.8%

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

   10. Simplified 67.8%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]
   11. Step-by-step derivation

       1. associate-*r/ 67.8%

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

   12. Applied egg-rr 67.8%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \mathsf{fma}\left(b, -1, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]
   13. Step-by-step derivation

       1. associate-/l* 67.9%

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

       2. associate-/r/ 67.8%

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

       3. fma-udef 67.8%

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

       4. *-commutative 67.8%

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

       5. neg-mul-1 67.8%

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

       6. +-commutative 67.8%

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

       7. unsub-neg 67.8%

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

   14. Simplified 67.8%

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

   if 6.99999999999999961e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.62e-74)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 7.2e-10)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7.2e-10) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.62d-74)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 7.2d-10) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7.2e-10) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.62e-74:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 7.2e-10:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.62e-74)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 7.2e-10)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.62e-74)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 7.2e-10)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.62e-74], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-10], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.62000000000000007e-74

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.6%

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

   4. Applied egg-rr 90.6%

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

   if -1.62000000000000007e-74 < b < 7.2e-10

   1. Initial program 74.0%

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

   2. Taylor expanded in b around 0 67.8%

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

   if 7.2e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.62e-74)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 7e-10)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7e-10) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.62d-74)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 7d-10) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.62e-74) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 7e-10) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.62e-74:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 7e-10:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.62e-74)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 7e-10)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.62e-74)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 7e-10)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.62e-74], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-10], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.62000000000000007e-74

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.6%

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

   4. Applied egg-rr 90.6%

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

   if -1.62000000000000007e-74 < b < 6.99999999999999961e-10

   1. Initial program 74.0%

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

   2. Taylor expanded in b around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. *-commutative 67.8%

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

      3. *-commutative 67.8%

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

      4. associate-*l* 68.0%

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

   4. Simplified 68.0%

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

   if 6.99999999999999961e-10 < b

   1. Initial program 8.0%

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

   2. Taylor expanded in b around inf 93.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 67.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.9%

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

   2. Taylor expanded in b around -inf 68.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 29.8%

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

   2. Taylor expanded in b around inf 67.9%

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

4. Final simplification 68.1%

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

Alternative 12: 67.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.9%

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

   2. Taylor expanded in b around -inf 68.3%

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

      1. associate-*r/ 68.4%

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

   4. Applied egg-rr 68.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 29.8%

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

   2. Taylor expanded in b around inf 67.9%

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

4. Final simplification 68.1%

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

Alternative 13: 67.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6e-267) (/ (* b -2.0) (* 3.0 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6d-267) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6e-267:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6e-267)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6e-267)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6e-267], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.9999999999999999e-267

   1. Initial program 75.8%

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

   2. Taylor expanded in b around -inf 65.6%

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

      1. *-commutative 65.6%

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

   4. Simplified 65.6%

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

   if 5.9999999999999999e-267 < b

   1. Initial program 26.9%

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

   2. Taylor expanded in b around inf 70.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14: 67.7% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6e-267) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6d-267) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6e-267:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6e-267)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6e-267)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6e-267], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.9999999999999999e-267

   1. Initial program 75.8%

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

   2. Taylor expanded in b around -inf 65.6%

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

      1. *-commutative 65.6%

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

   4. Simplified 65.6%

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

      1. div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l* 65.5%

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

      2. associate-/r* 65.5%

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

      3. metadata-eval 65.5%

         \[\leadsto b \cdot \left(-2 \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right) \]

      4. associate-*r/ 65.5%

         \[\leadsto b \cdot \color{blue}{\frac{-2 \cdot 0.3333333333333333}{a}} \]

      5. metadata-eval 65.5%

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

   8. Simplified 65.5%

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

   if 5.9999999999999999e-267 < b

   1. Initial program 26.9%

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

   2. Taylor expanded in b around inf 70.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 15: 67.7% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6e-267) (/ b (/ a -0.6666666666666666)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6d-267) then
        tmp = b / (a / (-0.6666666666666666d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6e-267:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6e-267)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6e-267)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6e-267], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.9999999999999999e-267

   1. Initial program 75.8%

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

      1. neg-sub0 75.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 75.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 75.8%

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

      4. neg-mul-1 75.8%

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

      5. associate-*r/ 75.8%

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

      6. *-commutative 75.8%

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

      7. metadata-eval 75.8%

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

      8. metadata-eval 75.8%

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

      9. times-frac 75.8%

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

      10. *-commutative 75.8%

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

      11. times-frac 75.7%

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

   3. Simplified 75.7%

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

      1. fma-udef 75.7%

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

      2. associate-*r* 75.6%

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

      3. *-commutative 75.6%

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

      4. metadata-eval 75.6%

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

      5. cancel-sign-sub-inv 75.6%

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

   5. Applied egg-rr 75.6%

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

   6. Taylor expanded in b around -inf 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r/ 65.6%

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

   8. Simplified 65.6%

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

   if 5.9999999999999999e-267 < b

   1. Initial program 26.9%

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

   2. Taylor expanded in b around inf 70.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 16: 67.7% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6e-267) (/ (* b -0.6666666666666666) a) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6d-267) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-267) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6e-267:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6e-267)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6e-267)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6e-267], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.9999999999999999e-267

   1. Initial program 75.8%

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

      1. associate-*r* 75.7%

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

      2. cancel-sign-sub-inv 75.7%

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

      3. metadata-eval 75.7%

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

      4. *-commutative 75.7%

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

      5. associate-*r* 75.7%

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

   3. Applied egg-rr 75.7%

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

   4. Taylor expanded in b around -inf 65.5%

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

      1. associate-*r/ 65.6%

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

      2. *-commutative 65.6%

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

   6. Simplified 65.6%

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

   if 5.9999999999999999e-267 < b

   1. Initial program 26.9%

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

   2. Taylor expanded in b around inf 70.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-267}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 17: 35.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

2. Taylor expanded in b around inf 34.2%

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

3. Final simplification 34.2%

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

Reproduce

herbie shell --seed 2023199 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))