Quadratic roots, full range

Percentage Accurate: 52.3% → 85.1%

Time: 10.5s

Alternatives: 6

Speedup: 19.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e+55)
   (- (/ c b) (/ b a))
   (if (<= b 1.15e-40)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-40) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.4d+55)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.15d-40) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-40) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e+55:
		tmp = (c / b) - (b / a)
	elif b <= 1.15e-40:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e+55)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.15e-40)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e+55)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.15e-40)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e+55], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-40], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.40000000000000021e55

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

   6. Simplified 96.0%

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

   if -4.40000000000000021e55 < b < 1.15e-40

   1. Initial program 77.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   if 1.15e-40 < b

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 15.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 90.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-68)
   (/ (- b) a)
   (if (<= b 1.35e-40)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-68) {
		tmp = -b / a;
	} else if (b <= 1.35e-40) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-68)) then
        tmp = -b / a
    else if (b <= 1.35d-40) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-68) {
		tmp = -b / a;
	} else if (b <= 1.35e-40) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-68:
		tmp = -b / a
	elif b <= 1.35e-40:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-68)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.35e-40)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-68)
		tmp = -b / a;
	elseif (b <= 1.35e-40)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-68], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.35e-40], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999998e-68

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 88.8%

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

      1. associate-*r/ 88.8%

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

      2. mul-1-neg 88.8%

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

   6. Simplified 88.8%

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

   if -1.14999999999999998e-68 < b < 1.35e-40

   1. Initial program 75.5%

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

      1. *-commutative 75.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 75.5%

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

      1. prod-diff 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}}{a \cdot 2} \]

      2. *-commutative 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{a \cdot 2} \]

      3. fma-def 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{a \cdot 2} \]

      4. associate-+l+ 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{a \cdot 2} \]

      5. pow2 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2}} + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      6. associate-*l* 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\left(-\color{blue}{4 \cdot \left(a \cdot c\right)}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      7. distribute-lft-neg-in 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      8. metadata-eval 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{-4} \cdot \left(a \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      9. *-commutative 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{\left(a \cdot c\right) \cdot -4} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      10. associate-*r* 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      11. *-commutative 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \mathsf{fma}\left(-c, 4 \cdot a, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)}}{a \cdot 2} \]

      12. fma-udef 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{\left(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(4 \cdot a\right) \cdot c\right)}\right)}}{a \cdot 2} \]

      13. distribute-lft-neg-in 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{\left(-c \cdot \left(4 \cdot a\right)\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      14. distribute-rgt-neg-in 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{c \cdot \left(-4 \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      15. distribute-lft-neg-in 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      16. metadata-eval 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(c \cdot \left(\color{blue}{-4} \cdot a\right) + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      17. associate-*l* 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{\left(c \cdot -4\right) \cdot a} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      18. *-commutative 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      19. *-commutative 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{c \cdot \left(4 \cdot a\right)}\right)\right)}}{a \cdot 2} \]

      20. associate-*r* 75.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{\left(c \cdot 4\right) \cdot a}\right)\right)}}{a \cdot 2} \]

   5. Applied egg-rr 75.1%

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

      1. +-commutative 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + \left(a \cdot \left(c \cdot -4\right) + \left(c \cdot 4\right) \cdot a\right)\right) + {b}^{2}}}}{a \cdot 2} \]

      2. associate-+r+ 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right) + \left(c \cdot 4\right) \cdot a\right)} + {b}^{2}}}{a \cdot 2} \]

      3. count-2 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \left(c \cdot 4\right) \cdot a\right) + {b}^{2}}}{a \cdot 2} \]

      4. associate-+l+ 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right) + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}}{a \cdot 2} \]

      5. count-2 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      6. associate-*r* 75.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\color{blue}{\left(a \cdot c\right) \cdot -4} + a \cdot \left(c \cdot -4\right)\right) + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      7. associate-*r* 75.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot -4}\right) + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      8. distribute-lft-out 75.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + -4\right)} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      9. metadata-eval 75.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-8} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      10. +-commutative 75.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left({b}^{2} + \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      11. unpow2 75.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \left(\color{blue}{b \cdot b} + \left(c \cdot 4\right) \cdot a\right)}}{a \cdot 2} \]

      12. fma-udef 75.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      13. *-commutative 75.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot 4\right)}\right)}}{a \cdot 2} \]

   7. Simplified 75.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)}}}{a \cdot 2} \]

   8. Taylor expanded in b around 0 65.9%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. mul-1-neg 65.9%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{a \cdot 2} \]

      2. unsub-neg 65.9%

         \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2} \]

      3. *-commutative 65.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2} \]

      4. *-commutative 65.9%

         \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left(a \cdot c\right) \cdot 4}} - b}{a \cdot 2} \]

      5. associate-*r* 65.9%

         \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{a \cdot \left(c \cdot 4\right)}} - b}{a \cdot 2} \]

      6. associate-*r* 66.0%

         \[\leadsto \frac{\sqrt{\color{blue}{a \cdot \left(c \cdot -8\right)} + a \cdot \left(c \cdot 4\right)} - b}{a \cdot 2} \]

      7. *-commutative 66.0%

         \[\leadsto \frac{\sqrt{a \cdot \color{blue}{\left(-8 \cdot c\right)} + a \cdot \left(c \cdot 4\right)} - b}{a \cdot 2} \]

      8. *-commutative 66.0%

         \[\leadsto \frac{\sqrt{a \cdot \left(-8 \cdot c\right) + a \cdot \color{blue}{\left(4 \cdot c\right)}} - b}{a \cdot 2} \]

      9. distribute-lft-in 66.4%

         \[\leadsto \frac{\sqrt{\color{blue}{a \cdot \left(-8 \cdot c + 4 \cdot c\right)}} - b}{a \cdot 2} \]

      10. distribute-rgt-out 66.4%

          \[\leadsto \frac{\sqrt{a \cdot \color{blue}{\left(c \cdot \left(-8 + 4\right)\right)}} - b}{a \cdot 2} \]

      11. metadata-eval 66.4%

          \[\leadsto \frac{\sqrt{a \cdot \left(c \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

   10. Simplified 66.4%

       \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}{a \cdot 2} \]

   if 1.35e-40 < b

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 15.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 90.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-119)
   (/ (- b) a)
   (if (<= b 1.1e-40) (/ (sqrt (* a (* c -4.0))) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-119) {
		tmp = -b / a;
	} else if (b <= 1.1e-40) {
		tmp = sqrt((a * (c * -4.0))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.5d-119)) then
        tmp = -b / a
    else if (b <= 1.1d-40) then
        tmp = sqrt((a * (c * (-4.0d0)))) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-119) {
		tmp = -b / a;
	} else if (b <= 1.1e-40) {
		tmp = Math.sqrt((a * (c * -4.0))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-119:
		tmp = -b / a
	elif b <= 1.1e-40:
		tmp = math.sqrt((a * (c * -4.0))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-119)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.1e-40)
		tmp = Float64(sqrt(Float64(a * Float64(c * -4.0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-119)
		tmp = -b / a;
	elseif (b <= 1.1e-40)
		tmp = sqrt((a * (c * -4.0))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-119], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.1e-40], N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5000000000000002e-119

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. mul-1-neg 84.8%

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

   6. Simplified 84.8%

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

   if -9.5000000000000002e-119 < b < 1.10000000000000004e-40

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 72.8%

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

      1. prod-diff 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}}{a \cdot 2} \]

      2. *-commutative 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{a \cdot 2} \]

      3. fma-def 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{a \cdot 2} \]

      4. associate-+l+ 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{a \cdot 2} \]

      5. pow2 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2}} + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      6. associate-*l* 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\left(-\color{blue}{4 \cdot \left(a \cdot c\right)}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      7. distribute-lft-neg-in 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      8. metadata-eval 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{-4} \cdot \left(a \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      9. *-commutative 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{\left(a \cdot c\right) \cdot -4} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      10. associate-*r* 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{a \cdot 2} \]

      11. *-commutative 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \mathsf{fma}\left(-c, 4 \cdot a, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)}}{a \cdot 2} \]

      12. fma-udef 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{\left(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(4 \cdot a\right) \cdot c\right)}\right)}}{a \cdot 2} \]

      13. distribute-lft-neg-in 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{\left(-c \cdot \left(4 \cdot a\right)\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      14. distribute-rgt-neg-in 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{c \cdot \left(-4 \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      15. distribute-lft-neg-in 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      16. metadata-eval 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(c \cdot \left(\color{blue}{-4} \cdot a\right) + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      17. associate-*l* 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{\left(c \cdot -4\right) \cdot a} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      18. *-commutative 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2} \]

      19. *-commutative 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{c \cdot \left(4 \cdot a\right)}\right)\right)}}{a \cdot 2} \]

      20. associate-*r* 72.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{b}^{2} + \left(a \cdot \left(c \cdot -4\right) + \left(a \cdot \left(c \cdot -4\right) + \color{blue}{\left(c \cdot 4\right) \cdot a}\right)\right)}}{a \cdot 2} \]

   5. Applied egg-rr 72.4%

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

      1. +-commutative 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + \left(a \cdot \left(c \cdot -4\right) + \left(c \cdot 4\right) \cdot a\right)\right) + {b}^{2}}}}{a \cdot 2} \]

      2. associate-+r+ 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right) + \left(c \cdot 4\right) \cdot a\right)} + {b}^{2}}}{a \cdot 2} \]

      3. count-2 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)} + \left(c \cdot 4\right) \cdot a\right) + {b}^{2}}}{a \cdot 2} \]

      4. associate-+l+ 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{2 \cdot \left(a \cdot \left(c \cdot -4\right)\right) + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}}{a \cdot 2} \]

      5. count-2 72.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + a \cdot \left(c \cdot -4\right)\right)} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      6. associate-*r* 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\color{blue}{\left(a \cdot c\right) \cdot -4} + a \cdot \left(c \cdot -4\right)\right) + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      7. associate-*r* 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot -4}\right) + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      8. distribute-lft-out 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + -4\right)} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      9. metadata-eval 72.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-8} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      10. +-commutative 72.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left({b}^{2} + \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      11. unpow2 72.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \left(\color{blue}{b \cdot b} + \left(c \cdot 4\right) \cdot a\right)}}{a \cdot 2} \]

      12. fma-udef 72.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      13. *-commutative 72.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot 4\right)}\right)}}{a \cdot 2} \]

   7. Simplified 72.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 71.9%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{\sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)}} \cdot \sqrt{\sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)}}}}{a \cdot 2} \]

      2. pow2 71.9%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\sqrt{\sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)}}\right)}^{2}}}{a \cdot 2} \]

      3. pow1/2 71.9%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)}^{0.5}}}\right)}^{2}}{a \cdot 2} \]

      4. sqrt-pow1 72.0%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a \cdot 2} \]

      5. associate-*l* 72.1%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{a \cdot \left(c \cdot -8\right)} + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      6. fma-def 72.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      7. metadata-eval 72.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

   9. Applied egg-rr 72.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]

   10. Taylor expanded in b around 0 67.5%

       \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. distribute-rgt-out 68.0%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}}}{a \cdot 2} \]

       2. metadata-eval 68.0%

          \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot \color{blue}{-4}}}{a \cdot 2} \]

       3. associate-*r* 68.0%

          \[\leadsto \frac{\sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

   12. Simplified 68.0%

       \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

   if 1.10000000000000004e-40 < b

   1. Initial program 15.6%

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

      1. *-commutative 15.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 15.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 90.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 67.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 71.0%

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

      1. +-commutative 71.0%

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

      2. mul-1-neg 71.0%

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

      3. unsub-neg 71.0%

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

   6. Simplified 71.0%

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

   if -1.000000000000002e-309 < b

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 29.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 70.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.1%

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

      2. distribute-neg-frac 70.1%

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

   6. Simplified 70.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 67.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-278) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-278) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.2d-278) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-278) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-278:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-278)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.2e-278)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-278], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1999999999999997e-278

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. mul-1-neg 70.5%

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

   6. Simplified 70.5%

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

   if 5.1999999999999997e-278 < b

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 29.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 70.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.7%

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

      2. distribute-neg-frac 70.7%

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

   6. Simplified 70.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 35.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. *-commutative 54.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]

4. Taylor expanded in b around -inf 40.3%

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

   1. associate-*r/ 40.3%

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

   2. mul-1-neg 40.3%

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

6. Simplified 40.3%

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

7. Final simplification 40.3%

   \[\leadsto \frac{-b}{a} \]

Reproduce

herbie shell --seed 2023320 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))