quadm (p42, negative)

Percentage Accurate: 52.4% → 83.6%

Time: 10.9s

Alternatives: 7

Speedup: 12.9×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{if}\;b \leq -1550000000000:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{c}{b}}}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))))
   (if (<= b -1550000000000.0)
     (/ c (- b))
     (if (<= b -1.35e-45)
       t_0
       (if (<= b -1.35e-99)
         (/ 1.0 (/ -1.0 (/ c b)))
         (if (<= b 8.6e+32) t_0 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	double tmp;
	if (b <= -1550000000000.0) {
		tmp = c / -b;
	} else if (b <= -1.35e-45) {
		tmp = t_0;
	} else if (b <= -1.35e-99) {
		tmp = 1.0 / (-1.0 / (c / b));
	} else if (b <= 8.6e+32) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    if (b <= (-1550000000000.0d0)) then
        tmp = c / -b
    else if (b <= (-1.35d-45)) then
        tmp = t_0
    else if (b <= (-1.35d-99)) then
        tmp = 1.0d0 / ((-1.0d0) / (c / b))
    else if (b <= 8.6d+32) then
        tmp = t_0
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	double tmp;
	if (b <= -1550000000000.0) {
		tmp = c / -b;
	} else if (b <= -1.35e-45) {
		tmp = t_0;
	} else if (b <= -1.35e-99) {
		tmp = 1.0 / (-1.0 / (c / b));
	} else if (b <= 8.6e+32) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	tmp = 0
	if b <= -1550000000000.0:
		tmp = c / -b
	elif b <= -1.35e-45:
		tmp = t_0
	elif b <= -1.35e-99:
		tmp = 1.0 / (-1.0 / (c / b))
	elif b <= 8.6e+32:
		tmp = t_0
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1550000000000.0)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.35e-45)
		tmp = t_0;
	elseif (b <= -1.35e-99)
		tmp = Float64(1.0 / Float64(-1.0 / Float64(c / b)));
	elseif (b <= 8.6e+32)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	tmp = 0.0;
	if (b <= -1550000000000.0)
		tmp = c / -b;
	elseif (b <= -1.35e-45)
		tmp = t_0;
	elseif (b <= -1.35e-99)
		tmp = 1.0 / (-1.0 / (c / b));
	elseif (b <= 8.6e+32)
		tmp = t_0;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1550000000000.0], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.35e-45], t$95$0, If[LessEqual[b, -1.35e-99], N[(1.0 / N[(-1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e+32], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.55e12

   1. Initial program 8.6%

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

      1. div-sub 7.1%

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

      2. sub-neg 7.1%

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

      3. neg-mul-1 7.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 7.1%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 6.9%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 6.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 6.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 6.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 7.1%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 8.7%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 8.7%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 8.7%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 8.7%

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

      14. *-commutative 8.7%

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

      15. *-commutative 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in b around -inf 92.6%

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

      1. associate-*r/ 92.6%

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

      2. neg-mul-1 92.6%

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

   7. Simplified 92.6%

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

   if -1.55e12 < b < -1.34999999999999992e-45 or -1.35e-99 < b < 8.5999999999999994e32

   1. Initial program 83.2%

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

   if -1.34999999999999992e-45 < b < -1.35e-99

   1. Initial program 27.3%

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

   3. Taylor expanded in b around -inf 60.3%

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

      1. clear-num 60.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{-2 \cdot \frac{a \cdot c}{b}}}} \]

      2. inv-pow 60.6%

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

      3. times-frac 60.6%

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

      4. metadata-eval 60.6%

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

      5. associate-/l* 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. associate-/r* 76.7%

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

      3. *-inverses 76.7%

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

      4. associate-*r/ 76.7%

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

      5. metadata-eval 76.7%

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

   7. Simplified 76.7%

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

   if 8.5999999999999994e32 < b

   1. Initial program 62.1%

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

      1. div-sub 62.1%

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

      2. sub-neg 62.1%

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

      3. neg-mul-1 62.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 62.1%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 62.1%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 62.1%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 62.1%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 62.1%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 62.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 62.0%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 62.0%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 62.0%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 62.0%

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

      14. *-commutative 62.0%

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

      15. *-commutative 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in a around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   7. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1550000000000:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{c}{b}}}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-126)
   (/ c (- b))
   (if (<= b 1.35e-97)
     (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-126) {
		tmp = c / -b;
	} else if (b <= 1.35e-97) {
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-1.5d-126)) then
        tmp = c / -b
    else if (b <= 1.35d-97) then
        tmp = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-126) {
		tmp = c / -b;
	} else if (b <= 1.35e-97) {
		tmp = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-126:
		tmp = c / -b
	elif b <= 1.35e-97:
		tmp = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-126)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.35e-97)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-126)
		tmp = c / -b;
	elseif (b <= 1.35e-97)
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-126], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.35e-97], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5000000000000001e-126

   1. Initial program 17.0%

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

      1. div-sub 15.8%

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

      2. sub-neg 15.8%

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

      3. neg-mul-1 15.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 15.8%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 15.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 15.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 15.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 15.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 15.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 17.0%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 17.0%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 17.0%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 17.0%

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

      14. *-commutative 17.0%

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

      15. *-commutative 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in b around -inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

   if -1.5000000000000001e-126 < b < 1.34999999999999993e-97

   1. Initial program 85.3%

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

      1. div-sub 85.3%

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

      2. sub-neg 85.3%

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

      3. neg-mul-1 85.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 85.3%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 85.3%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 85.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 85.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 85.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 85.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 85.2%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 85.2%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 85.2%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 85.2%

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

      14. *-commutative 85.2%

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

      15. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in b around 0 77.6%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 77.6%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}\right) \]

   7. Simplified 77.6%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}\right) \]

   if 1.34999999999999993e-97 < b

   1. Initial program 66.5%

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

      1. div-sub 66.5%

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

      2. sub-neg 66.5%

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

      3. neg-mul-1 66.5%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 66.5%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 66.4%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 66.4%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 66.4%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 66.4%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 66.4%

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

      14. *-commutative 66.4%

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

      15. *-commutative 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in a around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. mul-1-neg 90.6%

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

      3. unsub-neg 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-124}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-124)
   (/ c (- b))
   (if (<= b 6e-99)
     (/ (+ b (sqrt (* (* c a) -4.0))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-124) {
		tmp = c / -b;
	} else if (b <= 6e-99) {
		tmp = (b + sqrt(((c * a) * -4.0))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-124)) then
        tmp = c / -b
    else if (b <= 6d-99) then
        tmp = (b + sqrt(((c * a) * (-4.0d0)))) / (a * (-2.0d0))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-124) {
		tmp = c / -b;
	} else if (b <= 6e-99) {
		tmp = (b + Math.sqrt(((c * a) * -4.0))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-124:
		tmp = c / -b
	elif b <= 6e-99:
		tmp = (b + math.sqrt(((c * a) * -4.0))) / (a * -2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-124)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6e-99)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(c * a) * -4.0))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-124)
		tmp = c / -b;
	elseif (b <= 6e-99)
		tmp = (b + sqrt(((c * a) * -4.0))) / (a * -2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-124], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6e-99], N[(N[(b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.99999999999999973e-124

   1. Initial program 17.0%

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

      1. div-sub 15.8%

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

      2. sub-neg 15.8%

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

      3. neg-mul-1 15.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 15.8%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 15.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 15.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 15.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 15.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 15.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 17.0%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 17.0%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 17.0%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 17.0%

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

      14. *-commutative 17.0%

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

      15. *-commutative 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in b around -inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

   if -3.99999999999999973e-124 < b < 6.00000000000000012e-99

   1. Initial program 85.3%

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

      1. div-sub 85.3%

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

      2. sub-neg 85.3%

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

      3. neg-mul-1 85.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 85.3%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 85.3%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 85.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 85.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 85.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 85.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 85.2%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 85.2%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 85.2%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 85.2%

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

      14. *-commutative 85.2%

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

      15. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in b around 0 77.6%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 77.6%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}\right) \]

   7. Simplified 77.6%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}\right) \]
   8. Step-by-step derivation

      1. sqrt-prod 42.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{\sqrt{-4 \cdot a} \cdot \sqrt{c}}\right) \]

      2. *-commutative 42.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot -4}} \cdot \sqrt{c}\right) \]

   9. Applied egg-rr 42.2%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\right) \]
   10. Step-by-step derivation

       1. add0 42.2%

          \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot -4} \cdot \sqrt{c}\right) + 0} \]

       2. associate-*l/ 42.1%

          \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{a \cdot -4} \cdot \sqrt{c}\right)}{a}} + 0 \]

       3. *-un-lft-identity 42.1%

          \[\leadsto \frac{-0.5 \cdot \left(b + \sqrt{a \cdot -4} \cdot \sqrt{c}\right)}{\color{blue}{1 \cdot a}} + 0 \]

       4. times-frac 42.1%

          \[\leadsto \color{blue}{\frac{-0.5}{1} \cdot \frac{b + \sqrt{a \cdot -4} \cdot \sqrt{c}}{a}} + 0 \]

       5. metadata-eval 42.1%

          \[\leadsto \color{blue}{-0.5} \cdot \frac{b + \sqrt{a \cdot -4} \cdot \sqrt{c}}{a} + 0 \]

       6. *-commutative 42.1%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{c} \cdot \sqrt{a \cdot -4}}}{a} + 0 \]

       7. sqrt-unprod 77.6%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{a} + 0 \]

   11. Applied egg-rr 77.6%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a} + 0} \]
   12. Step-by-step derivation

       1. add0 77.6%

          \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}} \]

       2. metadata-eval 77.6%

          \[\leadsto \color{blue}{\frac{1}{-2}} \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \]

       3. times-frac 77.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{-2 \cdot a}} \]

       4. *-lft-identity 77.6%

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

       5. associate-*r* 77.6%

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

   13. Simplified 77.6%

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

   if 6.00000000000000012e-99 < b

   1. Initial program 66.5%

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

      1. div-sub 66.5%

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

      2. sub-neg 66.5%

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

      3. neg-mul-1 66.5%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 66.5%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 66.4%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 66.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 66.4%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 66.4%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 66.4%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 66.4%

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

      14. *-commutative 66.4%

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

      15. *-commutative 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in a around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. mul-1-neg 90.6%

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

      3. unsub-neg 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-124}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 29.8%

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

      1. div-sub 28.9%

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

      2. sub-neg 28.9%

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

      3. neg-mul-1 28.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 28.9%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 28.7%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 28.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 28.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 28.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 28.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 29.8%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 29.8%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 29.8%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 29.8%

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

      14. *-commutative 29.8%

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

      15. *-commutative 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in b around -inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   7. Simplified 70.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 71.9%

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

      1. div-sub 71.9%

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

      2. sub-neg 71.9%

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

      3. neg-mul-1 71.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 71.9%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 71.9%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 71.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 71.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 71.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 71.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 71.9%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 71.9%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 71.9%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 71.9%

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

      14. *-commutative 71.9%

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

      15. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. mul-1-neg 66.0%

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

      3. unsub-neg 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 68.3%

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

Alternative 5: 43.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+33}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1.45e+33) (/ c b) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e+33) {
		tmp = c / b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.45d+33)) then
        tmp = c / b
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e+33) {
		tmp = c / b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e+33:
		tmp = c / b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e+33)
		tmp = Float64(c / b);
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.45e+33)
		tmp = c / b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e+33], N[(c / b), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.45000000000000012e33

   1. Initial program 5.6%

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

      1. div-sub 3.8%

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

      2. sub-neg 3.8%

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

      3. neg-mul-1 3.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 3.8%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 3.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 3.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 3.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 3.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 3.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 5.6%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 5.6%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 5.6%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 5.6%

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

      14. *-commutative 5.6%

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

      15. *-commutative 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in b around inf 2.2%

      \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)} \]

   6. Taylor expanded in a around inf 22.6%

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

   if -1.45000000000000012e33 < b

   1. Initial program 69.9%

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

      1. div-sub 70.0%

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

      2. sub-neg 70.0%

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

      3. neg-mul-1 70.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 70.0%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 70.0%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 70.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 70.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 70.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 69.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 69.9%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 69.9%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 69.9%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 69.9%

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

      14. *-commutative 69.9%

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

      15. *-commutative 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in a around 0 46.6%

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

      1. associate-*r/ 46.6%

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

      2. mul-1-neg 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+33}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-231) (/ c (- b)) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-231) {
		tmp = c / -b;
	} else {
		tmp = b / -a;
	}
	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 <= (-2.2d-231)) then
        tmp = c / -b
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-231) {
		tmp = c / -b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-231:
		tmp = c / -b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-231)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e-231)
		tmp = c / -b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-231], N[(c / (-b)), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.20000000000000009e-231

   1. Initial program 24.8%

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

      1. div-sub 23.7%

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

      2. sub-neg 23.7%

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

      3. neg-mul-1 23.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 23.7%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 23.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 23.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 23.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 23.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 23.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 24.8%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 24.8%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 24.8%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 24.8%

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

      14. *-commutative 24.8%

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

      15. *-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in b around -inf 75.4%

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

      1. associate-*r/ 75.4%

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

      2. neg-mul-1 75.4%

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

   7. Simplified 75.4%

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

   if -2.20000000000000009e-231 < b

   1. Initial program 73.8%

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

      1. div-sub 73.8%

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

      2. sub-neg 73.8%

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

      3. neg-mul-1 73.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 73.8%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 73.8%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 73.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 73.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 73.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 73.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 73.7%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 73.7%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 73.7%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 73.7%

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

      14. *-commutative 73.7%

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

      15. *-commutative 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in a around 0 61.5%

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

      1. associate-*r/ 61.5%

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

      2. mul-1-neg 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 11.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. div-sub 49.6%

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

   2. sub-neg 49.6%

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

   3. neg-mul-1 49.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

   4. *-commutative 49.6%

      \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

   5. associate-/l* 49.5%

      \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

   6. distribute-neg-frac 49.5%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   7. neg-mul-1 49.5%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

   8. *-commutative 49.5%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

   9. associate-/l* 49.5%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

   10. distribute-rgt-out 50.0%

       \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

   11. associate-/r* 50.0%

       \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

   12. metadata-eval 50.0%

       \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

   13. fma-neg 50.0%

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

   14. *-commutative 50.0%

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

   15. *-commutative 50.0%

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

3. Simplified 50.0%

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

5. Taylor expanded in b around inf 32.3%

   \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)} \]

6. Taylor expanded in a around inf 9.0%

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

7. Final simplification 9.0%

   \[\leadsto \frac{c}{b} \]
8. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024046 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0))) (/ (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))