quadm (p42, negative)

Percentage Accurate: 52.2% → 85.4%

Time: 15.1s

Alternatives: 9

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: 52.2% 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: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-99}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{+98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-99)
   (- (/ c b))
   (if (<= b 1e+98)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-99) {
		tmp = -(c / b);
	} else if (b <= 1e+98) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (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 <= (-6d-99)) then
        tmp = -(c / b)
    else if (b <= 1d+98) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (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 <= -6e-99) {
		tmp = -(c / b);
	} else if (b <= 1e+98) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-99:
		tmp = -(c / b)
	elif b <= 1e+98:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-99)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1e+98)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / 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 <= -6e-99)
		tmp = -(c / b);
	elseif (b <= 1e+98)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-99], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1e+98], 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], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000012e-99

   1. Initial program 20.0%

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

      1. sub-neg 20.0%

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

      2. distribute-neg-out 20.0%

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

      3. neg-mul-1 20.0%

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

      4. times-frac 20.0%

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

      5. metadata-eval 20.0%

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

      6. sub-neg 20.0%

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

      7. +-commutative 20.0%

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

      8. *-commutative 20.0%

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

      9. distribute-lft-neg-in 20.0%

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

      10. distribute-rgt-neg-out 20.0%

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

      11. associate-*l* 20.0%

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

      12. fma-def 20.0%

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

      13. distribute-lft-neg-in 20.0%

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

      14. distribute-rgt-neg-in 20.0%

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

      15. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around -inf 84.9%

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

      1. mul-1-neg 84.9%

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

   7. Simplified 84.9%

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

   if -6.00000000000000012e-99 < b < 9.99999999999999998e97

   1. Initial program 81.4%

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

   if 9.99999999999999998e97 < b

   1. Initial program 47.8%

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

      1. sub-neg 47.8%

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

      2. distribute-neg-out 47.8%

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

      3. neg-mul-1 47.8%

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

      4. times-frac 47.8%

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

      5. metadata-eval 47.8%

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

      6. sub-neg 47.8%

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

      7. +-commutative 47.8%

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

      8. *-commutative 47.8%

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

      9. distribute-lft-neg-in 47.8%

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

      10. distribute-rgt-neg-out 47.8%

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

      11. associate-*l* 47.8%

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

      12. fma-def 48.0%

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

      13. distribute-lft-neg-in 48.0%

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

      14. distribute-rgt-neg-in 48.0%

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

      15. metadata-eval 48.0%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in b around inf 93.4%

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

      1. +-commutative 93.4%

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

      2. mul-1-neg 93.4%

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

      3. unsub-neg 93.4%

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

   7. Simplified 93.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-99}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{+98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-93}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-93)
   (- (/ c b))
   (if (<= b 4e-37)
     (/ (- (- b) (sqrt (* a (* c -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-93) {
		tmp = -(c / b);
	} else if (b <= 4e-37) {
		tmp = (-b - sqrt((a * (c * -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 <= (-2.3d-93)) then
        tmp = -(c / b)
    else if (b <= 4d-37) then
        tmp = (-b - sqrt((a * (c * (-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 <= -2.3e-93) {
		tmp = -(c / b);
	} else if (b <= 4e-37) {
		tmp = (-b - Math.sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-93:
		tmp = -(c / b)
	elif b <= 4e-37:
		tmp = (-b - math.sqrt((a * (c * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-93)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 4e-37)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -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 <= -2.3e-93)
		tmp = -(c / b);
	elseif (b <= 4e-37)
		tmp = (-b - sqrt((a * (c * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-93], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 4e-37], N[(N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $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 < -2.2999999999999998e-93

   1. Initial program 20.0%

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

      1. sub-neg 20.0%

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

      2. distribute-neg-out 20.0%

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

      3. neg-mul-1 20.0%

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

      4. times-frac 20.0%

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

      5. metadata-eval 20.0%

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

      6. sub-neg 20.0%

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

      7. +-commutative 20.0%

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

      8. *-commutative 20.0%

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

      9. distribute-lft-neg-in 20.0%

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

      10. distribute-rgt-neg-out 20.0%

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

      11. associate-*l* 20.0%

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

      12. fma-def 20.0%

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

      13. distribute-lft-neg-in 20.0%

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

      14. distribute-rgt-neg-in 20.0%

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

      15. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around -inf 84.9%

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

      1. mul-1-neg 84.9%

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

   7. Simplified 84.9%

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

   if -2.2999999999999998e-93 < b < 4.00000000000000027e-37

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. sqr-neg 77.7%

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

      3. *-commutative 77.7%

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

      4. sqr-neg 77.7%

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

      5. associate-*r* 77.7%

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

      6. *-commutative 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in b around 0 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-*r* 72.2%

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

   7. Simplified 72.2%

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

   if 4.00000000000000027e-37 < b

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. distribute-neg-out 61.8%

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

      3. neg-mul-1 61.8%

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

      4. times-frac 61.8%

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

      5. metadata-eval 61.8%

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

      6. sub-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. *-commutative 61.8%

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

      9. distribute-lft-neg-in 61.8%

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

      10. distribute-rgt-neg-out 61.8%

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

      11. associate-*l* 61.8%

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

      12. fma-def 61.9%

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

      13. distribute-lft-neg-in 61.9%

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

      14. distribute-rgt-neg-in 61.9%

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

      15. metadata-eval 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in b around inf 86.2%

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

      1. +-commutative 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

   7. Simplified 86.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-93}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-99}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-128}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-99)
   (- (/ c b))
   (if (<= b 1.02e-128)
     (* (/ 0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-99) {
		tmp = -(c / b);
	} else if (b <= 1.02e-128) {
		tmp = (0.5 / a) * (b - sqrt((a * (c * -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 <= (-6d-99)) then
        tmp = -(c / b)
    else if (b <= 1.02d-128) then
        tmp = (0.5d0 / a) * (b - sqrt((a * (c * (-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 <= -6e-99) {
		tmp = -(c / b);
	} else if (b <= 1.02e-128) {
		tmp = (0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-99:
		tmp = -(c / b)
	elif b <= 1.02e-128:
		tmp = (0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-99)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.02e-128)
		tmp = Float64(Float64(0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -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 <= -6e-99)
		tmp = -(c / b);
	elseif (b <= 1.02e-128)
		tmp = (0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-99], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.02e-128], N[(N[(0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -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 < -6.00000000000000012e-99

   1. Initial program 20.0%

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

      1. sub-neg 20.0%

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

      2. distribute-neg-out 20.0%

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

      3. neg-mul-1 20.0%

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

      4. times-frac 20.0%

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

      5. metadata-eval 20.0%

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

      6. sub-neg 20.0%

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

      7. +-commutative 20.0%

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

      8. *-commutative 20.0%

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

      9. distribute-lft-neg-in 20.0%

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

      10. distribute-rgt-neg-out 20.0%

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

      11. associate-*l* 20.0%

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

      12. fma-def 20.0%

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

      13. distribute-lft-neg-in 20.0%

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

      14. distribute-rgt-neg-in 20.0%

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

      15. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around -inf 84.9%

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

      1. mul-1-neg 84.9%

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

   7. Simplified 84.9%

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

   if -6.00000000000000012e-99 < b < 1.02e-128

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. sqr-neg 77.8%

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

      3. *-commutative 77.8%

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

      4. sqr-neg 77.8%

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

      5. associate-*r* 77.8%

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

      6. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in a around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-*r* 76.8%

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

   9. Simplified 76.8%

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

   if 1.02e-128 < b

   1. Initial program 63.9%

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

      1. sub-neg 63.9%

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

      2. distribute-neg-out 63.9%

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

      3. neg-mul-1 63.9%

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

      4. times-frac 63.9%

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

      5. metadata-eval 63.9%

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

      6. sub-neg 63.9%

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

      7. +-commutative 63.9%

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

      8. *-commutative 63.9%

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

      9. distribute-lft-neg-in 63.9%

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

      10. distribute-rgt-neg-out 63.9%

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

      11. associate-*l* 63.9%

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

      12. fma-def 64.0%

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

      13. distribute-lft-neg-in 64.0%

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

      14. distribute-rgt-neg-in 64.0%

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

      15. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in b around inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-99}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-128}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-97}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e-97)
   (- (/ c b))
   (if (<= b 8.5e-37)
     (/ -0.5 (/ a (+ b (sqrt (* a (* c -4.0))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-97) {
		tmp = -(c / b);
	} else if (b <= 8.5e-37) {
		tmp = -0.5 / (a / (b + sqrt((a * (c * -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 <= (-3.1d-97)) then
        tmp = -(c / b)
    else if (b <= 8.5d-37) then
        tmp = (-0.5d0) / (a / (b + sqrt((a * (c * (-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 <= -3.1e-97) {
		tmp = -(c / b);
	} else if (b <= 8.5e-37) {
		tmp = -0.5 / (a / (b + Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.1e-97:
		tmp = -(c / b)
	elif b <= 8.5e-37:
		tmp = -0.5 / (a / (b + math.sqrt((a * (c * -4.0)))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e-97)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 8.5e-37)
		tmp = Float64(-0.5 / Float64(a / Float64(b + sqrt(Float64(a * Float64(c * -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 <= -3.1e-97)
		tmp = -(c / b);
	elseif (b <= 8.5e-37)
		tmp = -0.5 / (a / (b + sqrt((a * (c * -4.0)))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.1e-97], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 8.5e-37], N[(-0.5 / N[(a / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $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 < -3.10000000000000002e-97

   1. Initial program 20.0%

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

      1. sub-neg 20.0%

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

      2. distribute-neg-out 20.0%

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

      3. neg-mul-1 20.0%

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

      4. times-frac 20.0%

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

      5. metadata-eval 20.0%

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

      6. sub-neg 20.0%

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

      7. +-commutative 20.0%

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

      8. *-commutative 20.0%

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

      9. distribute-lft-neg-in 20.0%

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

      10. distribute-rgt-neg-out 20.0%

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

      11. associate-*l* 20.0%

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

      12. fma-def 20.0%

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

      13. distribute-lft-neg-in 20.0%

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

      14. distribute-rgt-neg-in 20.0%

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

      15. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around -inf 84.9%

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

      1. mul-1-neg 84.9%

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

   7. Simplified 84.9%

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

   if -3.10000000000000002e-97 < b < 8.5000000000000007e-37

   1. Initial program 77.7%

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

      1. sub-neg 77.7%

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

      2. distribute-neg-out 77.7%

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

      3. neg-mul-1 77.7%

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

      4. times-frac 77.7%

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

      5. metadata-eval 77.7%

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

      6. sub-neg 77.7%

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

      7. +-commutative 77.7%

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

      8. *-commutative 77.7%

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

      9. distribute-lft-neg-in 77.7%

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

      10. distribute-rgt-neg-out 77.7%

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

      11. associate-*l* 77.7%

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

      12. fma-def 77.7%

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

      13. distribute-lft-neg-in 77.7%

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

      14. distribute-rgt-neg-in 77.7%

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

      15. metadata-eval 77.7%

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

   3. Simplified 77.7%

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

      1. clear-num 77.6%

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

      2. un-div-inv 77.6%

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

      3. pow2 77.6%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in a around inf 72.1%

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

      1. *-commutative 70.6%

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

      2. associate-*r* 70.6%

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

   9. Simplified 72.1%

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

   if 8.5000000000000007e-37 < b

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. distribute-neg-out 61.8%

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

      3. neg-mul-1 61.8%

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

      4. times-frac 61.8%

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

      5. metadata-eval 61.8%

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

      6. sub-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. *-commutative 61.8%

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

      9. distribute-lft-neg-in 61.8%

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

      10. distribute-rgt-neg-out 61.8%

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

      11. associate-*l* 61.8%

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

      12. fma-def 61.9%

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

      13. distribute-lft-neg-in 61.9%

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

      14. distribute-rgt-neg-in 61.9%

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

      15. metadata-eval 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in b around inf 86.2%

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

      1. +-commutative 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

   7. Simplified 86.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-97}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(-Float64(c / 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 <= -2e-310)
		tmp = -(c / b);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-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 < -1.999999999999994e-310

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. distribute-neg-out 33.8%

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

      3. neg-mul-1 33.8%

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

      4. times-frac 33.8%

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

      5. metadata-eval 33.8%

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

      6. sub-neg 33.8%

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

      7. +-commutative 33.8%

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

      8. *-commutative 33.8%

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

      9. distribute-lft-neg-in 33.8%

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

      10. distribute-rgt-neg-out 33.8%

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

      11. associate-*l* 33.8%

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

      12. fma-def 33.8%

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

      13. distribute-lft-neg-in 33.8%

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

      14. distribute-rgt-neg-in 33.8%

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

      15. metadata-eval 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b around -inf 65.1%

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

      1. mul-1-neg 65.1%

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

   7. Simplified 65.1%

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

   if -1.999999999999994e-310 < b

   1. Initial program 69.3%

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

      1. sub-neg 69.3%

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

      2. distribute-neg-out 69.3%

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

      3. neg-mul-1 69.3%

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

      4. times-frac 69.3%

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

      5. metadata-eval 69.3%

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

      6. sub-neg 69.3%

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

      7. +-commutative 69.3%

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

      8. *-commutative 69.3%

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

      9. distribute-lft-neg-in 69.3%

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

      10. distribute-rgt-neg-out 69.3%

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

      11. associate-*l* 69.3%

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

      12. fma-def 69.4%

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

      13. distribute-lft-neg-in 69.4%

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

      14. distribute-rgt-neg-in 69.4%

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

      15. metadata-eval 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b around inf 63.4%

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

      1. +-commutative 63.4%

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

      2. mul-1-neg 63.4%

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

      3. unsub-neg 63.4%

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

   7. Simplified 63.4%

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

4. Final simplification 64.3%

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

Alternative 6: 67.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{-307}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.36e-307) (- (/ c b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.36e-307) {
		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 <= (-1.36d-307)) 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 <= -1.36e-307) {
		tmp = -(c / b);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.36e-307:
		tmp = -(c / b)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.36e-307)
		tmp = Float64(-Float64(c / b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.36e-307)
		tmp = -(c / b);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.36e-307], (-N[(c / b), $MachinePrecision]), N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.36e-307

   1. Initial program 33.3%

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

      1. sub-neg 33.3%

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

      2. distribute-neg-out 33.3%

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

      3. neg-mul-1 33.3%

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

      4. times-frac 33.3%

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

      5. metadata-eval 33.3%

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

      6. sub-neg 33.3%

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

      7. +-commutative 33.3%

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

      8. *-commutative 33.3%

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

      9. distribute-lft-neg-in 33.3%

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

      10. distribute-rgt-neg-out 33.3%

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

      11. associate-*l* 33.3%

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

      12. fma-def 33.3%

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

      13. distribute-lft-neg-in 33.3%

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

      14. distribute-rgt-neg-in 33.3%

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

      15. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in b around -inf 65.6%

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

      1. mul-1-neg 65.6%

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

   7. Simplified 65.6%

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

   if -1.36e-307 < b

   1. Initial program 69.6%

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

      1. sub-neg 69.6%

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

      2. distribute-neg-out 69.6%

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

      3. neg-mul-1 69.6%

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

      4. times-frac 69.6%

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

      5. metadata-eval 69.6%

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

      6. sub-neg 69.6%

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

      7. +-commutative 69.6%

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

      8. *-commutative 69.6%

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

      9. distribute-lft-neg-in 69.6%

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

      10. distribute-rgt-neg-out 69.6%

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

      11. associate-*l* 69.6%

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

      12. fma-def 69.6%

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

      13. distribute-lft-neg-in 69.6%

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

      14. distribute-rgt-neg-in 69.6%

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

      15. metadata-eval 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. mul-1-neg 62.5%

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

   7. Simplified 62.5%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{-307}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.6% accurate, 29.0× 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(-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 51.1%

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

   1. sub-neg 51.1%

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

   2. distribute-neg-out 51.1%

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

   3. neg-mul-1 51.1%

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

   4. times-frac 51.1%

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

   5. metadata-eval 51.1%

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

   6. sub-neg 51.1%

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

   7. +-commutative 51.1%

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

   8. *-commutative 51.1%

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

   9. distribute-lft-neg-in 51.1%

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

   10. distribute-rgt-neg-out 51.1%

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

   11. associate-*l* 51.1%

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

   12. fma-def 51.2%

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

   13. distribute-lft-neg-in 51.2%

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

   14. distribute-rgt-neg-in 51.2%

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

   15. metadata-eval 51.2%

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

3. Simplified 51.2%

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

5. Taylor expanded in b around -inf 34.3%

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

   1. mul-1-neg 34.3%

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

7. Simplified 34.3%

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

8. Final simplification 34.3%

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

Alternative 8: 2.5% accurate, 38.7× 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(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 51.1%

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

   1. *-commutative 51.1%

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

   2. sqr-neg 51.1%

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

   3. *-commutative 51.1%

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

   4. sqr-neg 51.1%

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

   5. associate-*r* 51.1%

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

   6. *-commutative 51.1%

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

3. Simplified 51.1%

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

6. Applied egg-rr 34.2%

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

7. Taylor expanded in b around -inf 2.6%

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

8. Final simplification 2.6%

   \[\leadsto \frac{b}{a} \]
9. Add Preprocessing

Alternative 9: 10.8% 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 51.1%

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

   1. *-commutative 51.1%

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

   2. sqr-neg 51.1%

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

   3. *-commutative 51.1%

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

   4. sqr-neg 51.1%

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

   5. associate-*r* 51.1%

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

   6. *-commutative 51.1%

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

3. Simplified 51.1%

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

6. Applied egg-rr 34.2%

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

7. Taylor expanded in a around 0 12.3%

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

8. Final simplification 12.3%

   \[\leadsto \frac{c}{b} \]
9. 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 2024018 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (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)))