Cubic critical

Percentage Accurate: 50.7% → 85.6%

Time: 12.4s

Alternatives: 10

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: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+136}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e+136)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 6.8e-38)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+136) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6.8e-38) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.6d+136)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 6.8d-38) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+136) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6.8e-38) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e+136:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 6.8e-38:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e+136)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6.8e-38)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.6e+136)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 6.8e-38)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e+136], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-38], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.6e136

   1. Initial program 43.4%

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

   2. Taylor expanded in b around -inf 97.9%

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

   if -4.6e136 < b < 6.8000000000000004e-38

   1. Initial program 81.8%

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

   if 6.8000000000000004e-38 < b

   1. Initial program 12.6%

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

   2. Taylor expanded in b around inf 94.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+136}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+137}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+137)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 8.5e-38)
     (* -0.3333333333333333 (/ (- b (sqrt (- (* b b) (* 3.0 (* a c))))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+137) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.5e-38) {
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.6d+137)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 8.5d-38) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((b * b) - (3.0d0 * (a * c))))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+137) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.5e-38) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e+137:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 8.5e-38:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((b * b) - (3.0 * (a * c))))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+137)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 8.5e-38)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c))))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e+137)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 8.5e-38)
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+137], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-38], 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[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6e137

   1. Initial program 43.4%

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

   2. Taylor expanded in b around -inf 97.9%

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

   if -3.6e137 < b < 8.50000000000000046e-38

   1. Initial program 81.8%

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

      1. /-rgt-identity 81.8%

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

      2. metadata-eval 81.8%

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

      3. associate-/l* 81.8%

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

      4. associate-*r/ 81.8%

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

      5. *-commutative 81.8%

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

      6. associate-*l/ 81.8%

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

      7. associate-*r/ 81.8%

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

      8. metadata-eval 81.8%

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

      9. metadata-eval 81.8%

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

      10. times-frac 81.8%

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

      11. neg-mul-1 81.8%

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

      12. distribute-rgt-neg-in 81.8%

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

      13. times-frac 81.8%

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

      14. metadata-eval 81.8%

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

      15. neg-mul-1 81.8%

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

   3. Simplified 81.7%

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

      1. fma-udef 81.7%

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

      2. associate-*r* 81.7%

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

      3. *-commutative 81.7%

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

      4. metadata-eval 81.7%

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

      5. cancel-sign-sub-inv 81.7%

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

   5. Applied egg-rr 81.7%

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

   if 8.50000000000000046e-38 < b

   1. Initial program 12.6%

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

   2. Taylor expanded in b around inf 94.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+137}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+137)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2.2e-37)
     (* (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+137) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.2e-37) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+137)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2.2d-37) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+137) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.2e-37) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+137:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2.2e-37:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+137)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.2e-37)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+137)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2.2e-37)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) * (0.3333333333333333 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+137], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-37], 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[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e137

   1. Initial program 43.4%

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

   2. Taylor expanded in b around -inf 97.9%

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

   if -1e137 < b < 2.20000000000000002e-37

   1. Initial program 81.8%

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

      1. neg-sub0 81.8%

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

      2. associate-+l- 81.8%

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

      3. sub0-neg 81.8%

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

      4. neg-mul-1 81.8%

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

      5. associate-*r/ 81.8%

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

      6. *-commutative 81.8%

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

      7. metadata-eval 81.8%

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

      8. metadata-eval 81.8%

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

      9. times-frac 81.8%

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

      10. *-commutative 81.8%

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

      11. times-frac 81.8%

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

   3. Simplified 81.8%

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

      1. fma-udef 81.7%

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

      2. associate-*r* 81.7%

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

      3. *-commutative 81.7%

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

      4. metadata-eval 81.7%

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

      5. cancel-sign-sub-inv 81.7%

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

   5. Applied egg-rr 81.7%

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

   if 2.20000000000000002e-37 < b

   1. Initial program 12.6%

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

   2. Taylor expanded in b around inf 94.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-37}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-30)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 7.2e-45)
     (* -0.3333333333333333 (/ (- b (sqrt (* (* a c) -3.0))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 7.2e-45) {
		tmp = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.5d-30)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 7.2d-45) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((a * c) * (-3.0d0)))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 7.2e-45) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((a * c) * -3.0))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-30:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 7.2e-45:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((a * c) * -3.0))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-30)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 7.2e-45)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(a * c) * -3.0))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-30)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 7.2e-45)
		tmp = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-30], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-45], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5000000000000006e-30

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 91.7%

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

   if -7.5000000000000006e-30 < b < 7.20000000000000001e-45

   1. Initial program 75.0%

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

      1. /-rgt-identity 75.0%

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

      2. metadata-eval 75.0%

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

      3. associate-/l* 75.0%

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

      4. associate-*r/ 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-*l/ 75.0%

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

      7. associate-*r/ 75.0%

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

      8. metadata-eval 75.0%

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

      9. metadata-eval 75.0%

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

      10. times-frac 75.0%

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

      11. neg-mul-1 75.0%

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

      12. distribute-rgt-neg-in 75.0%

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

      13. times-frac 74.9%

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

      14. metadata-eval 74.9%

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

      15. neg-mul-1 74.9%

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

   3. Simplified 74.8%

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

      1. fma-udef 74.8%

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

      2. associate-*r* 74.8%

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

      3. *-commutative 74.8%

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

      4. metadata-eval 74.8%

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

      5. cancel-sign-sub-inv 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in b around 0 67.9%

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

   if 7.20000000000000001e-45 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-43}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.1e-30)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 5.2e-43)
     (* -0.3333333333333333 (/ (- b (sqrt (* a (* c -3.0)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.1e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.2e-43) {
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.1d-30)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 5.2d-43) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((a * (c * (-3.0d0))))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.1e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.2e-43) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.1e-30:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 5.2e-43:
		tmp = -0.3333333333333333 * ((b - math.sqrt((a * (c * -3.0)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.1e-30)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5.2e-43)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(a * Float64(c * -3.0)))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.1e-30)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 5.2e-43)
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.1e-30], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-43], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.09999999999999972e-30

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 91.7%

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

   if -5.09999999999999972e-30 < b < 5.2e-43

   1. Initial program 75.0%

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

      1. /-rgt-identity 75.0%

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

      2. metadata-eval 75.0%

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

      3. associate-/l* 75.0%

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

      4. associate-*r/ 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-*l/ 75.0%

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

      7. associate-*r/ 75.0%

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

      8. metadata-eval 75.0%

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

      9. metadata-eval 75.0%

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

      10. times-frac 75.0%

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

      11. neg-mul-1 75.0%

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

      12. distribute-rgt-neg-in 75.0%

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

      13. times-frac 74.9%

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

      14. metadata-eval 74.9%

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

      15. neg-mul-1 74.9%

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

   3. Simplified 74.8%

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

      1. fma-udef 74.8%

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

      2. associate-*r* 74.8%

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

      3. *-commutative 74.8%

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

      4. metadata-eval 74.8%

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

      5. cancel-sign-sub-inv 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in b around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*l* 68.0%

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

   8. Simplified 68.0%

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

   if 5.2e-43 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-43}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-45}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-30)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 9e-45)
     (* -0.3333333333333333 (/ (- b (sqrt (* c (* a -3.0)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 9e-45) {
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6d-30)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 9d-45) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((c * (a * (-3.0d0))))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 9e-45) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-30:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 9e-45:
		tmp = -0.3333333333333333 * ((b - math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-30)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 9e-45)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-30)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 9e-45)
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-30], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e-45], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999998e-30

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 91.7%

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

   if -5.9999999999999998e-30 < b < 8.9999999999999997e-45

   1. Initial program 75.0%

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

      1. /-rgt-identity 75.0%

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

      2. metadata-eval 75.0%

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

      3. associate-/l* 75.0%

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

      4. associate-*r/ 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-*l/ 75.0%

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

      7. associate-*r/ 75.0%

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

      8. metadata-eval 75.0%

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

      9. metadata-eval 75.0%

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

      10. times-frac 75.0%

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

      11. neg-mul-1 75.0%

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

      12. distribute-rgt-neg-in 75.0%

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

      13. times-frac 74.9%

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

      14. metadata-eval 74.9%

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

      15. neg-mul-1 74.9%

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

   3. Simplified 74.8%

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

      1. fma-udef 74.8%

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

      2. associate-*r* 74.8%

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

      3. *-commutative 74.8%

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

      4. metadata-eval 74.8%

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

      5. cancel-sign-sub-inv 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in b around 0 67.9%

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

      1. expm1-log1p-u 65.7%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-3 \cdot \left(c \cdot a\right)}\right)\right)}}{a} \]

      2. expm1-udef 37.7%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1\right)}}{a} \]

      3. *-commutative 37.7%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}\right)} - 1\right)}{a} \]

      4. associate-*r* 37.7%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}\right)} - 1\right)}{a} \]

   8. Applied egg-rr 37.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{c \cdot \left(a \cdot -3\right)}\right)} - 1\right)}}{a} \]
   9. Step-by-step derivation

      1. expm1-def 65.7%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{c \cdot \left(a \cdot -3\right)}\right)\right)}}{a} \]

      2. expm1-log1p 68.1%

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

   10. Simplified 68.1%

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

   if 8.9999999999999997e-45 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-45}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-30)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2.5e-41)
     (* (/ 0.3333333333333333 a) (- (sqrt (* c (* a -3.0))) b))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.5e-41) {
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.2d-30)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2.5d-41) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((c * (a * (-3.0d0)))) - b)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.5e-41) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((c * (a * -3.0))) - b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e-30:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2.5e-41:
		tmp = (0.3333333333333333 / a) * (math.sqrt((c * (a * -3.0))) - b)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-30)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.5e-41)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(c * Float64(a * -3.0))) - b));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e-30)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2.5e-41)
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-30], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-41], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.19999999999999973e-30

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 91.7%

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

   if -5.19999999999999973e-30 < b < 2.4999999999999998e-41

   1. Initial program 75.0%

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

      1. neg-sub0 75.0%

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

      2. associate-+l- 75.0%

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

      3. sub0-neg 75.0%

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

      4. neg-mul-1 75.0%

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

      5. associate-*r/ 75.0%

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

      6. *-commutative 75.0%

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

      7. metadata-eval 75.0%

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

      8. metadata-eval 75.0%

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

      9. times-frac 75.0%

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

      10. *-commutative 75.0%

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

      11. times-frac 74.9%

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

   3. Simplified 74.9%

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

      1. fma-udef 74.8%

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

      2. associate-*r* 74.8%

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

      3. *-commutative 74.8%

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

      4. metadata-eval 74.8%

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

      5. cancel-sign-sub-inv 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in b around 0 68.0%

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

      1. *-commutative 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*l* 68.0%

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

   8. Simplified 68.1%

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

      1. expm1-log1p-u 50.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]

      2. expm1-udef 21.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1} \]

      3. *-commutative 21.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)}\right)} - 1 \]

   10. Applied egg-rr 21.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 50.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\right)\right)} \]

       2. expm1-log1p 68.1%

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

       3. *-commutative 68.1%

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

       4. associate-*r* 68.1%

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

       5. *-commutative 68.1%

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

   12. Simplified 68.1%

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

   if 2.4999999999999998e-41 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-30)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 6.2e-45)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6.2e-45) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.5d-30)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 6.2d-45) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-30) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6.2e-45) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-30:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 6.2e-45:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-30)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6.2e-45)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-30)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 6.2e-45)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-30], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-45], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5000000000000006e-30

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 91.7%

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

   if -7.5000000000000006e-30 < b < 6.2000000000000002e-45

   1. Initial program 75.0%

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

   2. Taylor expanded in b around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. *-commutative 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*l* 68.1%

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

   4. Simplified 68.1%

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

   if 6.2000000000000002e-45 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 9: 67.6% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 73.4%

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

   2. Taylor expanded in b around -inf 68.8%

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

      1. *-commutative 68.8%

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

   4. Simplified 68.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.2%

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

   2. Taylor expanded in b around inf 71.1%

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

4. Final simplification 69.9%

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

Alternative 10: 36.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

2. Taylor expanded in b around inf 34.3%

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

3. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))