Cubic critical

Percentage Accurate: 52.6% → 85.8%

Time: 24.3s

Alternatives: 13

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{+119}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.05e+119)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 6.5e-108)
     (/ (- (sqrt (fma c (* a -3.0) (pow b 2.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.05e+119) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 6.5e-108) {
		tmp = (sqrt(fma(c, (a * -3.0), pow(b, 2.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.05e+119)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 6.5e-108)
		tmp = Float64(Float64(sqrt(fma(c, Float64(a * -3.0), (b ^ 2.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.05e+119], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-108], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[Power[b, 2.0], $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 < -3.05e119

   1. Initial program 38.0%

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

   3. Taylor expanded in b around 0 38.0%

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

      1. cancel-sign-sub-inv 38.0%

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

      2. metadata-eval 38.0%

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

      3. associate-*r* 38.0%

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

      4. *-commutative 38.0%

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

      5. *-commutative 38.0%

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

      6. +-commutative 38.0%

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

      7. fma-define 38.1%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in b around -inf 90.3%

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

      1. *-commutative 90.3%

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

      2. associate-*l/ 90.1%

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

      3. associate-/l* 90.3%

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

   8. Simplified 90.3%

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

   if -3.05e119 < b < 6.5000000000000002e-108

   1. Initial program 78.9%

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

   3. Taylor expanded in b around 0 78.8%

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

      1. cancel-sign-sub-inv 78.8%

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

      2. metadata-eval 78.8%

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

      3. associate-*r* 78.9%

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

      4. *-commutative 78.9%

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

      5. *-commutative 78.9%

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

      6. +-commutative 78.9%

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

      7. fma-define 78.9%

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

   5. Simplified 78.9%

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

   if 6.5000000000000002e-108 < b

   1. Initial program 16.8%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{+119}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+119}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+119)
   (* b (/ -0.6666666666666666 a))
   (if (<= b 6.2e-108)
     (/ (- (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 <= -2.8e+119) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 6.2e-108) {
		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 <= (-2.8d+119)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 6.2d-108) 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 <= -2.8e+119) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 6.2e-108) {
		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 <= -2.8e+119:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 6.2e-108:
		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 <= -2.8e+119)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 6.2e-108)
		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 <= -2.8e+119)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 6.2e-108)
		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, -2.8e+119], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-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[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.80000000000000013e119

   1. Initial program 38.0%

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

   3. Taylor expanded in b around 0 38.0%

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

      1. cancel-sign-sub-inv 38.0%

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

      2. metadata-eval 38.0%

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

      3. associate-*r* 38.0%

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

      4. *-commutative 38.0%

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

      5. *-commutative 38.0%

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

      6. +-commutative 38.0%

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

      7. fma-define 38.1%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in b around -inf 90.3%

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

      1. *-commutative 90.3%

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

      2. associate-*l/ 90.1%

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

      3. associate-/l* 90.3%

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

   8. Simplified 90.3%

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

   if -2.80000000000000013e119 < b < 6.20000000000000028e-108

   1. Initial program 78.9%

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

   if 6.20000000000000028e-108 < b

   1. Initial program 16.8%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+119}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-108}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-155)
   (/ b (* a -1.5))
   (if (<= b 5.5e-108)
     (* 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 <= -1.35e-155) {
		tmp = b / (a * -1.5);
	} else if (b <= 5.5e-108) {
		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 <= (-1.35d-155)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 5.5d-108) 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 <= -1.35e-155) {
		tmp = b / (a * -1.5);
	} else if (b <= 5.5e-108) {
		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 <= -1.35e-155:
		tmp = b / (a * -1.5)
	elif b <= 5.5e-108:
		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 <= -1.35e-155)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 5.5e-108)
		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 <= -1.35e-155)
		tmp = b / (a * -1.5);
	elseif (b <= 5.5e-108)
		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, -1.35e-155], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-108], 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 < -1.34999999999999991e-155

   1. Initial program 69.3%

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

   3. Taylor expanded in b around 0 69.3%

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

      1. cancel-sign-sub-inv 69.3%

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

      2. metadata-eval 69.3%

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

      3. associate-*r* 69.3%

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

      4. *-commutative 69.3%

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

      5. *-commutative 69.3%

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

      6. +-commutative 69.3%

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

      7. fma-define 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in b around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*l/ 77.8%

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

      3. associate-/l* 77.8%

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

   8. Simplified 77.8%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.9%

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

      3. div-inv 77.9%

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

      4. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

   if -1.34999999999999991e-155 < b < 5.50000000000000031e-108

   1. Initial program 65.8%

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

   3. Taylor expanded in b around 0 65.6%

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

      1. cancel-sign-sub-inv 65.6%

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

      2. metadata-eval 65.6%

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

      3. associate-*r* 65.8%

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

      4. *-commutative 65.8%

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

      5. *-commutative 65.8%

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

      6. +-commutative 65.8%

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

      7. fma-define 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in c around inf 65.6%

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

      1. *-commutative 65.6%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. associate-*r* 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 65.6%

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

      8. *-commutative 65.6%

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

   8. Simplified 65.6%

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

      1. div-inv 65.4%

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

      2. add-sqr-sqrt 24.2%

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

      3. sqrt-unprod 64.9%

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

      4. sqr-neg 64.9%

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

      5. sqrt-unprod 41.6%

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

      6. add-sqr-sqrt 65.0%

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

      7. *-commutative 65.0%

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

   10. Applied egg-rr 65.0%

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

       1. associate-*r/ 65.2%

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

       2. *-commutative 65.2%

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

       3. *-commutative 65.2%

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

       4. times-frac 65.1%

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

       5. metadata-eval 65.1%

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

   12. Simplified 65.1%

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

   if 5.50000000000000031e-108 < b

   1. Initial program 16.8%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-108}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-99}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.42e-99)
   (/ b (* a -1.5))
   (if (<= b 1.2e-109)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.42e-99) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.2e-109) {
		tmp = (sqrt((a * (c * -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 <= (-1.42d-99)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.2d-109) then
        tmp = (sqrt((a * (c * (-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 <= -1.42e-99) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.2e-109) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.42e-99:
		tmp = b / (a * -1.5)
	elif b <= 1.2e-109:
		tmp = (math.sqrt((a * (c * -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 <= -1.42e-99)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.2e-109)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -1.42e-99)
		tmp = b / (a * -1.5);
	elseif (b <= 1.2e-109)
		tmp = (sqrt((a * (c * -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, -1.42e-99], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e-109], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -1.42e-99

   1. Initial program 68.8%

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

   3. Taylor expanded in b around 0 68.8%

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

      1. cancel-sign-sub-inv 68.8%

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

      2. metadata-eval 68.8%

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

      3. associate-*r* 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

      6. +-commutative 68.8%

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

      7. fma-define 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in b around -inf 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l/ 81.3%

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

      3. associate-/l* 81.4%

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

   8. Simplified 81.4%

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

      1. clear-num 81.2%

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

      2. un-div-inv 81.4%

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

      3. div-inv 81.5%

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

      4. metadata-eval 81.5%

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

   10. Applied egg-rr 81.5%

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

   if -1.42e-99 < b < 1.19999999999999994e-109

   1. Initial program 66.9%

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

      1. prod-diff 66.2%

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

      2. distribute-rgt-neg-in 66.2%

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

      3. *-commutative 66.2%

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

      4. distribute-rgt-neg-in 66.2%

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

      5. metadata-eval 66.2%

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

      6. *-commutative 66.2%

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

      7. fma-undefine 66.2%

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

      8. distribute-lft-neg-in 66.2%

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

      9. distribute-rgt-neg-in 66.2%

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

      10. *-commutative 66.2%

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

      11. distribute-rgt-neg-in 66.2%

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

      12. metadata-eval 66.2%

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

      13. *-commutative 66.2%

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

      14. associate-*l* 66.1%

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

   4. Applied egg-rr 66.1%

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

   5. Taylor expanded in b around 0 62.4%

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

      1. distribute-rgt-out 63.0%

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

      2. metadata-eval 63.0%

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

      3. associate-*r* 63.0%

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

      4. mul-1-neg 63.0%

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

      5. unsub-neg 63.0%

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

   7. Simplified 63.0%

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

   if 1.19999999999999994e-109 < b

   1. Initial program 16.8%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-99}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-100}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-100)
   (/ b (* a -1.5))
   (if (<= b 1.25e-109)
     (/ (- (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 <= -4.4e-100) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-109) {
		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 <= (-4.4d-100)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.25d-109) 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 <= -4.4e-100) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-109) {
		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 <= -4.4e-100:
		tmp = b / (a * -1.5)
	elif b <= 1.25e-109:
		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 <= -4.4e-100)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.25e-109)
		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 <= -4.4e-100)
		tmp = b / (a * -1.5);
	elseif (b <= 1.25e-109)
		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, -4.4e-100], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-109], 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 < -4.39999999999999978e-100

   1. Initial program 68.8%

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

   3. Taylor expanded in b around 0 68.8%

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

      1. cancel-sign-sub-inv 68.8%

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

      2. metadata-eval 68.8%

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

      3. associate-*r* 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

      6. +-commutative 68.8%

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

      7. fma-define 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in b around -inf 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l/ 81.3%

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

      3. associate-/l* 81.4%

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

   8. Simplified 81.4%

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

      1. clear-num 81.2%

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

      2. un-div-inv 81.4%

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

      3. div-inv 81.5%

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

      4. metadata-eval 81.5%

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

   10. Applied egg-rr 81.5%

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

   if -4.39999999999999978e-100 < b < 1.25000000000000005e-109

   1. Initial program 66.9%

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

   3. Taylor expanded in b around 0 63.0%

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

      1. associate-*r* 63.1%

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

      2. *-commutative 63.1%

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

      3. *-commutative 63.1%

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

   5. Simplified 63.1%

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

   if 1.25000000000000005e-109 < b

   1. Initial program 16.8%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-100}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-155}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-155)
   (/ b (* a -1.5))
   (if (<= b 6.8e-108)
     (* (sqrt (* a (* c -3.0))) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-155) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.8e-108) {
		tmp = sqrt((a * (c * -3.0))) * (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 <= (-1.25d-155)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 6.8d-108) then
        tmp = sqrt((a * (c * (-3.0d0)))) * (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 <= -1.25e-155) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.8e-108) {
		tmp = Math.sqrt((a * (c * -3.0))) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-155:
		tmp = b / (a * -1.5)
	elif b <= 6.8e-108:
		tmp = math.sqrt((a * (c * -3.0))) * (0.3333333333333333 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-155)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6.8e-108)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) * 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 <= -1.25e-155)
		tmp = b / (a * -1.5);
	elseif (b <= 6.8e-108)
		tmp = sqrt((a * (c * -3.0))) * (0.3333333333333333 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-155], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-108], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $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 < -1.25e-155

   1. Initial program 69.3%

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

   3. Taylor expanded in b around 0 69.3%

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

      1. cancel-sign-sub-inv 69.3%

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

      2. metadata-eval 69.3%

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

      3. associate-*r* 69.3%

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

      4. *-commutative 69.3%

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

      5. *-commutative 69.3%

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

      6. +-commutative 69.3%

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

      7. fma-define 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in b around -inf 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*l/ 77.8%

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

      3. associate-/l* 77.8%

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

   8. Simplified 77.8%

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

      1. clear-num 77.7%

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

      2. un-div-inv 77.9%

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

      3. div-inv 77.9%

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

      4. metadata-eval 77.9%

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

   10. Applied egg-rr 77.9%

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

   if -1.25e-155 < b < 6.80000000000000004e-108

   1. Initial program 65.8%

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

      1. prod-diff 65.1%

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

      2. distribute-rgt-neg-in 65.1%

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

      3. *-commutative 65.1%

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

      4. distribute-rgt-neg-in 65.1%

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

      5. metadata-eval 65.1%

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

      6. *-commutative 65.1%

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

      7. fma-undefine 65.1%

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

      8. distribute-lft-neg-in 65.1%

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

      9. distribute-rgt-neg-in 65.1%

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

      10. *-commutative 65.1%

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

      11. distribute-rgt-neg-in 65.1%

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

      12. metadata-eval 65.1%

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

      13. *-commutative 65.1%

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

      14. associate-*l* 65.0%

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

   4. Applied egg-rr 65.0%

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

   5. Taylor expanded in b around 0 64.2%

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

      1. associate-*r* 64.1%

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

      2. associate-*r/ 64.3%

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

      3. metadata-eval 64.3%

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

      4. distribute-rgt-out 65.0%

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

      5. metadata-eval 65.0%

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

      6. associate-*r* 65.0%

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

   7. Simplified 65.0%

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

   if 6.80000000000000004e-108 < b

   1. Initial program 16.8%

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

   3. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

   5. Simplified 90.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-155}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{c \cdot \frac{-0.3333333333333333}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-110)
   (/ b (* a -1.5))
   (if (<= b 9.5e-133)
     (sqrt (* c (/ -0.3333333333333333 a)))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-110) {
		tmp = b / (a * -1.5);
	} else if (b <= 9.5e-133) {
		tmp = sqrt((c * (-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 <= (-3.7d-110)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 9.5d-133) then
        tmp = sqrt((c * ((-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 <= -3.7e-110) {
		tmp = b / (a * -1.5);
	} else if (b <= 9.5e-133) {
		tmp = Math.sqrt((c * (-0.3333333333333333 / a)));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-110:
		tmp = b / (a * -1.5)
	elif b <= 9.5e-133:
		tmp = math.sqrt((c * (-0.3333333333333333 / a)))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-110)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 9.5e-133)
		tmp = sqrt(Float64(c * 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 <= -3.7e-110)
		tmp = b / (a * -1.5);
	elseif (b <= 9.5e-133)
		tmp = sqrt((c * (-0.3333333333333333 / a)));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-110], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-133], N[Sqrt[N[(c * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.70000000000000016e-110

   1. Initial program 68.8%

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

   3. Taylor expanded in b around 0 68.8%

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

      1. cancel-sign-sub-inv 68.8%

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

      2. metadata-eval 68.8%

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

      3. associate-*r* 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

      6. +-commutative 68.8%

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

      7. fma-define 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in b around -inf 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l/ 81.3%

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

      3. associate-/l* 81.4%

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

   8. Simplified 81.4%

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

      1. clear-num 81.2%

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

      2. un-div-inv 81.4%

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

      3. div-inv 81.5%

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

      4. metadata-eval 81.5%

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

   10. Applied egg-rr 81.5%

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

   if -3.70000000000000016e-110 < b < 9.4999999999999992e-133

   1. Initial program 66.8%

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

      1. prod-diff 66.0%

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

      2. distribute-rgt-neg-in 66.0%

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

      3. *-commutative 66.0%

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

      4. distribute-rgt-neg-in 66.0%

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

      5. metadata-eval 66.0%

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

      6. *-commutative 66.0%

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

      7. fma-undefine 66.0%

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

      8. distribute-lft-neg-in 66.0%

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

      9. distribute-rgt-neg-in 66.0%

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

      10. *-commutative 66.0%

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

      11. distribute-rgt-neg-in 66.0%

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

      12. metadata-eval 66.0%

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

      13. *-commutative 66.0%

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

      14. associate-*l* 65.9%

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

   4. Applied egg-rr 65.9%

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

   5. Taylor expanded in a around inf 38.0%

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

      1. *-commutative 38.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-6 \cdot c + 3 \cdot c}{a}} \cdot 0.3333333333333333} \]

      2. distribute-rgt-out 38.0%

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

      3. metadata-eval 38.0%

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

   7. Simplified 38.0%

      \[\leadsto \color{blue}{\sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 37.9%

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

      2. sqrt-unprod 38.0%

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

      3. swap-sqr 38.0%

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

      4. add-sqr-sqrt 38.2%

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

      5. associate-/l* 38.1%

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

      6. metadata-eval 38.1%

         \[\leadsto \sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot \color{blue}{0.1111111111111111}} \]

   9. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot 0.1111111111111111}} \]
   10. Step-by-step derivation

       1. associate-*l* 38.1%

          \[\leadsto \sqrt{\color{blue}{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}} \]

       2. associate-*l/ 38.2%

          \[\leadsto \sqrt{c \cdot \color{blue}{\frac{-3 \cdot 0.1111111111111111}{a}}} \]

       3. metadata-eval 38.2%

          \[\leadsto \sqrt{c \cdot \frac{\color{blue}{-0.3333333333333333}}{a}} \]

   11. Simplified 38.2%

       \[\leadsto \color{blue}{\sqrt{c \cdot \frac{-0.3333333333333333}{a}}} \]

   if 9.4999999999999992e-133 < b

   1. Initial program 19.5%

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

   3. Taylor expanded in b around inf 87.2%

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

      1. *-commutative 87.2%

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

   5. Simplified 87.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{c \cdot \frac{-0.3333333333333333}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-308) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.4d-308) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-308:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-308)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-308)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-308], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000008e-308

   1. Initial program 67.6%

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

   3. Taylor expanded in b around 0 67.6%

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

      1. cancel-sign-sub-inv 67.6%

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

      2. metadata-eval 67.6%

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

      3. associate-*r* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

      6. +-commutative 67.6%

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

      7. fma-define 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in b around -inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l/ 63.7%

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

      3. associate-/l* 63.7%

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

   8. Simplified 63.7%

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

   if 2.40000000000000008e-308 < b

   1. Initial program 29.0%

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

   3. Taylor expanded in c around 0 66.0%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

      2. metadata-eval 66.0%

         \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{\color{blue}{0.5}}{b}\right) \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)} \]

   6. Taylor expanded in c around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. associate-*l/ 72.1%

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

      3. *-commutative 72.1%

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

   8. Simplified 72.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-308) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		tmp = b * (-0.6666666666666666 / 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 <= 2.4d-308) then
        tmp = b * ((-0.6666666666666666d0) / 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 <= 2.4e-308) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-308:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-308)
		tmp = Float64(b * Float64(-0.6666666666666666 / 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 <= 2.4e-308)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-308], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000008e-308

   1. Initial program 67.6%

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

   3. Taylor expanded in b around 0 67.6%

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

      1. cancel-sign-sub-inv 67.6%

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

      2. metadata-eval 67.6%

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

      3. associate-*r* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

      6. +-commutative 67.6%

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

      7. fma-define 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in b around -inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l/ 63.7%

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

      3. associate-/l* 63.7%

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

   8. Simplified 63.7%

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

   if 2.40000000000000008e-308 < b

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf 72.3%

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

      1. *-commutative 72.3%

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

   5. Simplified 72.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-308) (/ -0.6666666666666666 (/ a b)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		tmp = -0.6666666666666666 / (a / 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 <= 2.4d-308) then
        tmp = (-0.6666666666666666d0) / (a / 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 <= 2.4e-308) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-308:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-308)
		tmp = Float64(-0.6666666666666666 / Float64(a / 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 <= 2.4e-308)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-308], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000008e-308

   1. Initial program 67.6%

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

   3. Taylor expanded in b around 0 67.6%

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

      1. cancel-sign-sub-inv 67.6%

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

      2. metadata-eval 67.6%

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

      3. associate-*r* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

      6. +-commutative 67.6%

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

      7. fma-define 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in b around -inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l/ 63.7%

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

      3. associate-/l* 63.7%

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

   8. Simplified 63.7%

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

   9. Taylor expanded in b around 0 63.7%

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

       1. associate-*r/ 63.7%

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

       2. associate-*l/ 63.7%

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

       3. metadata-eval 63.7%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{a} \cdot b \]

       4. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot a}} \cdot b \]

       5. *-commutative 63.7%

          \[\leadsto \frac{1}{\color{blue}{a \cdot -1.5}} \cdot b \]

       6. associate-/r/ 63.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -1.5}{b}}} \]

       7. associate-*l/ 63.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       8. *-commutative 63.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       9. associate-/r* 63.8%

          \[\leadsto \color{blue}{\frac{\frac{1}{-1.5}}{\frac{a}{b}}} \]

       10. metadata-eval 63.8%

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

   11. Simplified 63.8%

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

   if 2.40000000000000008e-308 < b

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf 72.3%

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

      1. *-commutative 72.3%

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

   5. Simplified 72.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-308) (/ b (* a -1.5)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.4d-308) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-308) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-308:
		tmp = b / (a * -1.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-308)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-308)
		tmp = b / (a * -1.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-308], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000008e-308

   1. Initial program 67.6%

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

   3. Taylor expanded in b around 0 67.6%

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

      1. cancel-sign-sub-inv 67.6%

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

      2. metadata-eval 67.6%

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

      3. associate-*r* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

      6. +-commutative 67.6%

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

      7. fma-define 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in b around -inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l/ 63.7%

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

      3. associate-/l* 63.7%

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

   8. Simplified 63.7%

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

      1. clear-num 63.6%

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

      2. un-div-inv 63.8%

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

      3. div-inv 63.8%

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

      4. metadata-eval 63.8%

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

   10. Applied egg-rr 63.8%

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

   if 2.40000000000000008e-308 < b

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf 72.3%

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

      1. *-commutative 72.3%

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

   5. Simplified 72.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 2.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* 1.3333333333333333 (/ b a)))

C

double code(double a, double b, double c) {
	return 1.3333333333333333 * (b / 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 = 1.3333333333333333d0 * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return 1.3333333333333333 * (b / a);
}

Python

def code(a, b, c):
	return 1.3333333333333333 * (b / a)

Julia

function code(a, b, c)
	return Float64(1.3333333333333333 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.3333333333333333 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(1.3333333333333333 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.8%

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

3. Simplified 46.7%

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

   1. *-un-lft-identity 46.7%

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

   2. *-un-lft-identity 46.7%

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

   3. prod-diff 46.7%

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

6. Applied egg-rr 23.3%

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

   1. associate-+l+ 23.3%

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

   2. fma-undefine 23.3%

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

   3. *-rgt-identity 23.3%

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

8. Simplified 23.3%

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

9. Taylor expanded in b around inf 2.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]

10. Final simplification 2.4%

    \[\leadsto 1.3333333333333333 \cdot \frac{b}{a} \]
11. Add Preprocessing

Alternative 13: 36.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.8%

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

3. Taylor expanded in b around 0 46.8%

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

   1. cancel-sign-sub-inv 46.8%

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

   2. metadata-eval 46.8%

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

   3. associate-*r* 46.8%

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

   4. *-commutative 46.8%

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

   5. *-commutative 46.8%

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

   6. +-commutative 46.8%

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

   7. fma-define 46.8%

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

5. Simplified 46.8%

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

6. Taylor expanded in b around -inf 30.8%

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

   1. *-commutative 30.8%

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

   2. associate-*l/ 30.8%

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

   3. associate-/l* 30.9%

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

8. Simplified 30.9%

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

9. Final simplification 30.9%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))