Cubic critical

Percentage Accurate: 52.4% → 85.8%

Time: 11.7s

Alternatives: 11

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: 52.4% 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: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e+153)
   (/ (+ (* b -2.0) (* (/ (* a 1.5) b) c)) (* a 3.0))
   (if (<= b 1.7e-108)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e+153) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 1.7e-108) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} 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 <= (-4.2d+153)) then
        tmp = ((b * (-2.0d0)) + (((a * 1.5d0) / b) * c)) / (a * 3.0d0)
    else if (b <= 1.7d-108) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    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 <= -4.2e+153) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 1.7e-108) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e+153:
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0)
	elif b <= 1.7e-108:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e+153)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(a * 1.5) / b) * c)) / Float64(a * 3.0));
	elseif (b <= 1.7e-108)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	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 <= -4.2e+153)
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	elseif (b <= 1.7e-108)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e+153], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-108], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.20000000000000033e153

   1. Initial program 37.1%

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

      1. sqr-neg 37.1%

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

      2. sqr-neg 37.1%

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

      3. associate-*l* 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in b around -inf 92.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 92.0%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 92.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   6. Simplified 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. fma-udef 99.7%

         \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{1.5 \cdot a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 99.7%

         \[\leadsto \frac{b \cdot -2 + \color{blue}{\frac{1.5 \cdot a}{b} \cdot c}}{3 \cdot a} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{b \cdot -2 + \frac{\color{blue}{a \cdot 1.5}}{b} \cdot c}{3 \cdot a} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}}{3 \cdot a} \]

   if -4.20000000000000033e153 < b < 1.70000000000000001e-108

   1. Initial program 84.4%

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

   if 1.70000000000000001e-108 < b

   1. Initial program 22.6%

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

      1. sqr-neg 22.6%

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

      2. sqr-neg 22.6%

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

      3. associate-*l* 22.6%

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

   3. Simplified 22.6%

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

   4. Taylor expanded in b around inf 83.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+153)
   (/ (+ (* b -2.0) (* (/ (* a 1.5) b) c)) (* a 3.0))
   (if (<= b 2e-108)
     (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (/ a 0.3333333333333333))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+153) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 2e-108) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a / 0.3333333333333333);
	} 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 <= (-4d+153)) then
        tmp = ((b * (-2.0d0)) + (((a * 1.5d0) / b) * c)) / (a * 3.0d0)
    else if (b <= 2d-108) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a / 0.3333333333333333d0)
    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 <= -4e+153) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 2e-108) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a / 0.3333333333333333);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+153:
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0)
	elif b <= 2e-108:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a / 0.3333333333333333)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+153)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(a * 1.5) / b) * c)) / Float64(a * 3.0));
	elseif (b <= 2e-108)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a / 0.3333333333333333));
	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 <= -4e+153)
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	elseif (b <= 2e-108)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a / 0.3333333333333333);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+153], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-108], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4e153

   1. Initial program 37.1%

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

      1. sqr-neg 37.1%

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

      2. sqr-neg 37.1%

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

      3. associate-*l* 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in b around -inf 92.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 92.0%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 92.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   6. Simplified 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. fma-udef 99.7%

         \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{1.5 \cdot a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 99.7%

         \[\leadsto \frac{b \cdot -2 + \color{blue}{\frac{1.5 \cdot a}{b} \cdot c}}{3 \cdot a} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{b \cdot -2 + \frac{\color{blue}{a \cdot 1.5}}{b} \cdot c}{3 \cdot a} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}}{3 \cdot a} \]

   if -4e153 < b < 2.00000000000000008e-108

   1. Initial program 84.4%

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

      1. neg-sub0 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-+l- 84.4%

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

      4. sub0-neg 84.4%

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

      5. neg-mul-1 84.4%

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

   3. Simplified 84.1%

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

      1. associate-*r* 84.1%

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

      2. metadata-eval 84.1%

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

      3. distribute-rgt-neg-in 84.1%

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

      4. *-commutative 84.1%

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

      5. fma-neg 84.1%

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

      6. associate-*r* 84.1%

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

      7. *-commutative 84.1%

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

      8. associate-*l* 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. *-commutative 84.1%

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

   7. Simplified 84.1%

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

   if 2.00000000000000008e-108 < b

   1. Initial program 22.6%

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

      1. sqr-neg 22.6%

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

      2. sqr-neg 22.6%

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

      3. associate-*l* 22.6%

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

   3. Simplified 22.6%

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

   4. Taylor expanded in b around inf 83.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+152)
   (/ (+ (* b -2.0) (* (/ (* a 1.5) b) c)) (* a 3.0))
   (if (<= b 1.45e-108)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+152) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 1.45e-108) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} 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+152)) then
        tmp = ((b * (-2.0d0)) + (((a * 1.5d0) / b) * c)) / (a * 3.0d0)
    else if (b <= 1.45d-108) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    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+152) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 1.45e-108) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+152:
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0)
	elif b <= 1.45e-108:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+152)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(a * 1.5) / b) * c)) / Float64(a * 3.0));
	elseif (b <= 1.45e-108)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	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+152)
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	elseif (b <= 1.45e-108)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+152], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-108], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5e152

   1. Initial program 37.1%

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

      1. sqr-neg 37.1%

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

      2. sqr-neg 37.1%

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

      3. associate-*l* 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in b around -inf 92.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 92.0%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 92.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   6. Simplified 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. fma-udef 99.7%

         \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{1.5 \cdot a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 99.7%

         \[\leadsto \frac{b \cdot -2 + \color{blue}{\frac{1.5 \cdot a}{b} \cdot c}}{3 \cdot a} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{b \cdot -2 + \frac{\color{blue}{a \cdot 1.5}}{b} \cdot c}{3 \cdot a} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}}{3 \cdot a} \]

   if -5e152 < b < 1.45e-108

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.2%

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

   3. Simplified 84.2%

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

   if 1.45e-108 < b

   1. Initial program 22.6%

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

      1. sqr-neg 22.6%

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

      2. sqr-neg 22.6%

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

      3. associate-*l* 22.6%

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

   3. Simplified 22.6%

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

   4. Taylor expanded in b around inf 83.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-26}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.35e-26)
   (/ (+ (* b -2.0) (* (/ (* a 1.5) b) c)) (* a 3.0))
   (if (<= b 2e-108)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.35e-26) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 2e-108) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} 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 <= (-2.35d-26)) then
        tmp = ((b * (-2.0d0)) + (((a * 1.5d0) / b) * c)) / (a * 3.0d0)
    else if (b <= 2d-108) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    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 <= -2.35e-26) {
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	} else if (b <= 2e-108) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.35e-26:
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0)
	elif b <= 2e-108:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.35e-26)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(a * 1.5) / b) * c)) / Float64(a * 3.0));
	elseif (b <= 2e-108)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	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 <= -2.35e-26)
		tmp = ((b * -2.0) + (((a * 1.5) / b) * c)) / (a * 3.0);
	elseif (b <= 2e-108)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.35e-26], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-108], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.34999999999999995e-26

   1. Initial program 63.4%

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

      1. sqr-neg 63.4%

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

      2. sqr-neg 63.4%

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

      3. associate-*l* 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in b around -inf 87.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 87.2%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 87.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 91.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

      4. associate-*r/ 91.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}}\right)}{3 \cdot a} \]

   6. Simplified 91.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot a}{\frac{b}{c}}\right)}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. fma-udef 91.6%

         \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{1.5 \cdot a}{\frac{b}{c}}}}{3 \cdot a} \]

      2. associate-/r/ 91.6%

         \[\leadsto \frac{b \cdot -2 + \color{blue}{\frac{1.5 \cdot a}{b} \cdot c}}{3 \cdot a} \]

      3. *-commutative 91.6%

         \[\leadsto \frac{b \cdot -2 + \frac{\color{blue}{a \cdot 1.5}}{b} \cdot c}{3 \cdot a} \]

   8. Applied egg-rr 91.6%

      \[\leadsto \frac{\color{blue}{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}}{3 \cdot a} \]

   if -2.34999999999999995e-26 < b < 2.00000000000000008e-108

   1. Initial program 77.5%

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

      1. sqr-neg 77.5%

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

      2. sqr-neg 77.5%

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

      3. associate-*l* 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in b around 0 75.1%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*l* 75.1%

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

   6. Simplified 75.1%

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

   if 2.00000000000000008e-108 < b

   1. Initial program 22.6%

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

      1. sqr-neg 22.6%

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

      2. sqr-neg 22.6%

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

      3. associate-*l* 22.6%

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

   3. Simplified 22.6%

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

   4. Taylor expanded in b around inf 83.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-26}:\\ \;\;\;\;\frac{b \cdot -2 + \frac{a \cdot 1.5}{b} \cdot c}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 67.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{a \cdot \frac{c \cdot -1.5}{b} - b \cdot -2}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (/ (- (* a (/ (* c -1.5) b)) (* b -2.0)) (* a -3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((a * ((c * -1.5) / b)) - (b * -2.0)) / (a * -3.0);
	} 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 = ((a * ((c * (-1.5d0)) / b)) - (b * (-2.0d0))) / (a * (-3.0d0))
    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 = ((a * ((c * -1.5) / b)) - (b * -2.0)) / (a * -3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((a * ((c * -1.5) / b)) - (b * -2.0)) / (a * -3.0)
	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(a * Float64(Float64(c * -1.5) / b)) - Float64(b * -2.0)) / Float64(a * -3.0));
	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 = ((a * ((c * -1.5) / b)) - (b * -2.0)) / (a * -3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(a * N[(N[(c * -1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * -3.0), $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 66.0%

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

      1. sqr-neg 66.0%

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

      2. sqr-neg 66.0%

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

      3. associate-*l* 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in b around -inf 64.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. associate-*r/ 64.2%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]

   6. Applied egg-rr 64.2%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\frac{1.5 \cdot \left(a \cdot c\right)}{b}}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. frac-2neg 64.2%

         \[\leadsto \color{blue}{\frac{-\left(-2 \cdot b + \frac{1.5 \cdot \left(a \cdot c\right)}{b}\right)}{-3 \cdot a}} \]

      2. div-inv 64.1%

         \[\leadsto \color{blue}{\left(-\left(-2 \cdot b + \frac{1.5 \cdot \left(a \cdot c\right)}{b}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

      3. *-commutative 64.1%

         \[\leadsto \left(-\left(\color{blue}{b \cdot -2} + \frac{1.5 \cdot \left(a \cdot c\right)}{b}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      4. fma-def 64.1%

         \[\leadsto \left(-\color{blue}{\mathsf{fma}\left(b, -2, \frac{1.5 \cdot \left(a \cdot c\right)}{b}\right)}\right) \cdot \frac{1}{-3 \cdot a} \]

      5. *-un-lft-identity 64.1%

         \[\leadsto \left(-\mathsf{fma}\left(b, -2, \frac{1.5 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot b}}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      6. times-frac 64.1%

         \[\leadsto \left(-\mathsf{fma}\left(b, -2, \color{blue}{\frac{1.5}{1} \cdot \frac{a \cdot c}{b}}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      7. metadata-eval 64.1%

         \[\leadsto \left(-\mathsf{fma}\left(b, -2, \color{blue}{1.5} \cdot \frac{a \cdot c}{b}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      8. associate-*r/ 67.2%

         \[\leadsto \left(-\mathsf{fma}\left(b, -2, 1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      9. *-commutative 67.2%

         \[\leadsto \left(-\mathsf{fma}\left(b, -2, 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)\right) \cdot \frac{1}{-\color{blue}{a \cdot 3}} \]

   8. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(b, -2, 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)\right) \cdot \frac{1}{-a \cdot 3}} \]
   9. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \color{blue}{\frac{\left(-\mathsf{fma}\left(b, -2, 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)\right) \cdot 1}{-a \cdot 3}} \]

      2. *-rgt-identity 67.2%

         \[\leadsto \frac{\color{blue}{-\mathsf{fma}\left(b, -2, 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{-a \cdot 3} \]

      3. neg-sub0 67.2%

         \[\leadsto \frac{\color{blue}{0 - \mathsf{fma}\left(b, -2, 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{-a \cdot 3} \]

      4. fma-udef 67.2%

         \[\leadsto \frac{0 - \color{blue}{\left(b \cdot -2 + 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{-a \cdot 3} \]

      5. *-commutative 67.2%

         \[\leadsto \frac{0 - \left(\color{blue}{-2 \cdot b} + 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{-a \cdot 3} \]

      6. associate-*r/ 64.2%

         \[\leadsto \frac{0 - \left(-2 \cdot b + 1.5 \cdot \color{blue}{\frac{a \cdot c}{b}}\right)}{-a \cdot 3} \]

      7. +-commutative 64.2%

         \[\leadsto \frac{0 - \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -2 \cdot b\right)}}{-a \cdot 3} \]

      8. associate-*r/ 67.2%

         \[\leadsto \frac{0 - \left(1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)} + -2 \cdot b\right)}{-a \cdot 3} \]

      9. *-commutative 67.2%

         \[\leadsto \frac{0 - \left(1.5 \cdot \left(a \cdot \frac{c}{b}\right) + \color{blue}{b \cdot -2}\right)}{-a \cdot 3} \]

      10. associate--r+ 67.2%

          \[\leadsto \frac{\color{blue}{\left(0 - 1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b \cdot -2}}{-a \cdot 3} \]

      11. neg-sub0 67.2%

          \[\leadsto \frac{\color{blue}{\left(-1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)} - b \cdot -2}{-a \cdot 3} \]

      12. distribute-lft-neg-in 67.2%

          \[\leadsto \frac{\color{blue}{\left(-1.5\right) \cdot \left(a \cdot \frac{c}{b}\right)} - b \cdot -2}{-a \cdot 3} \]

      13. metadata-eval 67.2%

          \[\leadsto \frac{\color{blue}{-1.5} \cdot \left(a \cdot \frac{c}{b}\right) - b \cdot -2}{-a \cdot 3} \]

      14. *-commutative 67.2%

          \[\leadsto \frac{\color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot -1.5} - b \cdot -2}{-a \cdot 3} \]

      15. associate-*l* 67.2%

          \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c}{b} \cdot -1.5\right)} - b \cdot -2}{-a \cdot 3} \]

      16. associate-*l/ 67.2%

          \[\leadsto \frac{a \cdot \color{blue}{\frac{c \cdot -1.5}{b}} - b \cdot -2}{-a \cdot 3} \]

      17. distribute-rgt-neg-in 67.2%

          \[\leadsto \frac{a \cdot \frac{c \cdot -1.5}{b} - b \cdot -2}{\color{blue}{a \cdot \left(-3\right)}} \]

      18. metadata-eval 67.2%

          \[\leadsto \frac{a \cdot \frac{c \cdot -1.5}{b} - b \cdot -2}{a \cdot \color{blue}{-3}} \]

   10. Simplified 67.2%

       \[\leadsto \color{blue}{\frac{a \cdot \frac{c \cdot -1.5}{b} - b \cdot -2}{a \cdot -3}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 39.1%

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

      1. sqr-neg 39.1%

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

      2. sqr-neg 39.1%

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

      3. associate-*l* 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in b around inf 65.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{a \cdot \frac{c \cdot -1.5}{b} - b \cdot -2}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 67.4% 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 66.0%

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

      1. sqr-neg 66.0%

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

      2. sqr-neg 66.0%

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

      3. associate-*l* 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in b around -inf 67.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 39.1%

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

      1. sqr-neg 39.1%

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

      2. sqr-neg 39.1%

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

      3. associate-*l* 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in b around inf 65.9%

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

4. Final simplification 66.5%

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

Alternative 7: 67.4% 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 66.0%

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

      1. sqr-neg 66.0%

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

      2. sqr-neg 66.0%

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

      3. associate-*l* 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in b around -inf 67.2%

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

      1. associate-*r/ 67.2%

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

   6. Applied egg-rr 67.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 39.1%

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

      1. sqr-neg 39.1%

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

      2. sqr-neg 39.1%

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

      3. associate-*l* 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in b around inf 65.9%

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

4. Final simplification 66.5%

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

Alternative 8: 67.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-297}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.2e-297) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.2e-297) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9.2d-297) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.2e-297) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.2e-297:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.2e-297)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.2e-297)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.2e-297], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.1999999999999996e-297

   1. Initial program 66.3%

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

      1. sqr-neg 66.3%

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

      2. sqr-neg 66.3%

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

      3. associate-*l* 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in b around -inf 66.4%

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

      1. *-commutative 66.4%

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

   6. Simplified 66.4%

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

   if 9.1999999999999996e-297 < b

   1. Initial program 38.6%

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

      1. sqr-neg 38.6%

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

      2. sqr-neg 38.6%

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

      3. associate-*l* 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in b around inf 66.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-297}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 67.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.5e-297) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.5e-297) {
		tmp = b / (a * -1.5);
	} 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 <= 5.5d-297) then
        tmp = b / (a * (-1.5d0))
    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 <= 5.5e-297) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.5e-297:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.5e-297)
		tmp = Float64(b / Float64(a * -1.5));
	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 <= 5.5e-297)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.5e-297], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5000000000000003e-297

   1. Initial program 66.3%

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

      1. sqr-neg 66.3%

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

      2. sqr-neg 66.3%

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

      3. associate-*l* 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in b around -inf 66.4%

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

      1. *-commutative 66.4%

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

   6. Simplified 66.4%

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

      1. associate-/r/ 66.4%

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

      2. div-inv 66.5%

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

      3. metadata-eval 66.5%

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

   8. Applied egg-rr 66.5%

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

   if 5.5000000000000003e-297 < b

   1. Initial program 38.6%

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

      1. sqr-neg 38.6%

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

      2. sqr-neg 38.6%

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

      3. associate-*l* 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in b around inf 66.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 34.7% 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 52.5%

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

   1. sqr-neg 52.5%

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

   2. sqr-neg 52.5%

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

   3. associate-*l* 52.4%

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

3. Simplified 52.4%

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

4. Taylor expanded in b around inf 34.3%

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

5. Final simplification 34.3%

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

Alternative 11: 11.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

   1. sqr-neg 52.5%

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

   2. sqr-neg 52.5%

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

   3. associate-*l* 52.4%

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

3. Simplified 52.4%

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

4. Taylor expanded in b around 0 52.4%

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

   1. *-commutative 52.4%

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

   2. unpow2 52.4%

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

   3. fma-def 52.4%

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

6. Simplified 52.4%

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

   1. div-inv 52.4%

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

   2. neg-mul-1 52.4%

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

   3. fma-def 52.4%

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

   4. *-commutative 52.4%

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

8. Applied egg-rr 52.4%

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

9. Taylor expanded in a around 0 11.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 11.2%

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

    2. distribute-rgt1-in 11.2%

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

    3. metadata-eval 11.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 11.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 11.2%

       \[\leadsto \frac{\color{blue}{0}}{a} \]

11. Simplified 11.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]

12. Final simplification 11.2%

    \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023268 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))