quadm (p42, negative)

Percentage Accurate: 52.3% → 85.4%

Time: 16.9s

Alternatives: 10

Speedup: 19.1×

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: 52.3% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-85}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e-85)
   (/ (- c) b)
   (if (<= b 4.5e+69)
     (fma -0.5 (/ b a) (/ (sqrt (fma a (* c -4.0) (* b b))) (* a -2.0)))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-85) {
		tmp = -c / b;
	} else if (b <= 4.5e+69) {
		tmp = fma(-0.5, (b / a), (sqrt(fma(a, (c * -4.0), (b * b))) / (a * -2.0)));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e-85)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4.5e+69)
		tmp = fma(-0.5, Float64(b / a), Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) / Float64(a * -2.0)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e-85], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4.5e+69], N[(-0.5 * N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4500000000000001e-85

   1. Initial program 22.9%

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

   2. Taylor expanded in b around -inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   4. Simplified 86.9%

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

   if -1.4500000000000001e-85 < b < 4.4999999999999999e69

   1. Initial program 82.0%

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

      1. div-sub 82.0%

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

      2. neg-mul-1 82.0%

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

      3. times-frac 82.0%

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

      4. fma-neg 82.0%

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

      5. metadata-eval 82.0%

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

      6. frac-2neg 82.0%

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

      7. distribute-neg-frac 82.0%

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

      8. remove-double-neg 82.0%

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

      9. fma-neg 81.9%

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

      10. distribute-lft-neg-in 81.9%

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

      11. *-commutative 81.9%

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

      12. associate-*l* 81.9%

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

      13. metadata-eval 81.9%

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

      14. *-commutative 81.9%

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

   3. Applied egg-rr 81.9%

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

      1. fma-def 82.0%

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

      2. +-commutative 82.0%

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

      3. fma-def 82.0%

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

   5. Simplified 82.0%

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

   if 4.4999999999999999e69 < b

   1. Initial program 57.8%

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

   2. Taylor expanded in b around inf 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-85}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-88)
   (/ (- c) b)
   (if (<= b 6e+69)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-88) {
		tmp = -c / b;
	} else if (b <= 6e+69) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} 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 <= (-7.6d-88)) then
        tmp = -c / b
    else if (b <= 6d+69) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    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 <= -7.6e-88) {
		tmp = -c / b;
	} else if (b <= 6e+69) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.6e-88:
		tmp = -c / b
	elif b <= 6e+69:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-88)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 6e+69)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.6e-88)
		tmp = -c / b;
	elseif (b <= 6e+69)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-88], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 6e+69], 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[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.60000000000000022e-88

   1. Initial program 22.9%

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

   2. Taylor expanded in b around -inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   4. Simplified 86.9%

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

   if -7.60000000000000022e-88 < b < 5.99999999999999967e69

   1. Initial program 82.0%

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

   if 5.99999999999999967e69 < b

   1. Initial program 57.8%

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

   2. Taylor expanded in b around inf 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-87)
   (/ (- c) b)
   (if (<= b 5.2e+67)
     (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* a (* c -4.0))))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-87) {
		tmp = -c / b;
	} else if (b <= 5.2e+67) {
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (a * (c * -4.0)))));
	} 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 <= (-3.8d-87)) then
        tmp = -c / b
    else if (b <= 5.2d+67) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((b * b) + (a * (c * (-4.0d0))))))
    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 <= -3.8e-87) {
		tmp = -c / b;
	} else if (b <= 5.2e+67) {
		tmp = (-0.5 / a) * (b + Math.sqrt(((b * b) + (a * (c * -4.0)))));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-87:
		tmp = -c / b
	elif b <= 5.2e+67:
		tmp = (-0.5 / a) * (b + math.sqrt(((b * b) + (a * (c * -4.0)))))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-87)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5.2e+67)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-87)
		tmp = -c / b;
	elseif (b <= 5.2e+67)
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (a * (c * -4.0)))));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-87], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5.2e+67], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-87

   1. Initial program 22.9%

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

   2. Taylor expanded in b around -inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   4. Simplified 86.9%

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

   if -3.8e-87 < b < 5.2000000000000001e67

   1. Initial program 82.0%

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

      1. frac-2neg 82.0%

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

      2. div-inv 81.8%

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

      3. *-commutative 81.8%

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

      4. distribute-lft-neg-in 81.8%

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

      5. associate-/r* 81.8%

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

      6. metadata-eval 81.8%

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

      7. metadata-eval 81.8%

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

      8. sub-neg 81.8%

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

      9. distribute-neg-out 81.8%

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

      10. remove-double-neg 81.8%

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

      11. fma-neg 81.8%

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

      12. distribute-lft-neg-in 81.8%

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

      13. *-commutative 81.8%

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

      14. associate-*l* 81.8%

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

      15. metadata-eval 81.8%

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

   3. Applied egg-rr 81.8%

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

      1. fma-udef 81.8%

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

   5. Applied egg-rr 81.8%

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

   if 5.2000000000000001e67 < b

   1. Initial program 57.8%

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

   2. Taylor expanded in b around inf 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-86}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-86)
   (/ (- c) b)
   (if (<= b 2.25e-120)
     (/ (- (- 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 <= -6e-86) {
		tmp = -c / b;
	} else if (b <= 2.25e-120) {
		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 <= (-6d-86)) then
        tmp = -c / b
    else if (b <= 2.25d-120) 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 <= -6e-86) {
		tmp = -c / b;
	} else if (b <= 2.25e-120) {
		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 <= -6e-86:
		tmp = -c / b
	elif b <= 2.25e-120:
		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 <= -6e-86)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.25e-120)
		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 <= -6e-86)
		tmp = -c / b;
	elseif (b <= 2.25e-120)
		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, -6e-86], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.25e-120], 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 < -6.0000000000000002e-86

   1. Initial program 22.9%

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

   2. Taylor expanded in b around -inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   4. Simplified 86.9%

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

   if -6.0000000000000002e-86 < b < 2.25e-120

   1. Initial program 79.4%

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

   2. Taylor expanded in b around 0 74.6%

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

      1. *-commutative 74.6%

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

      2. associate-*r* 74.6%

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

   4. Simplified 74.6%

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

   if 2.25e-120 < b

   1. Initial program 67.2%

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

   2. Taylor expanded in b around inf 88.3%

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

      1. +-commutative 88.3%

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

      2. mul-1-neg 88.3%

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

      3. unsub-neg 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-86}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-120}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-88)
   (/ (- c) b)
   (if (<= b 2.8e-120)
     (* (/ -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 <= -3.2e-88) {
		tmp = -c / b;
	} else if (b <= 2.8e-120) {
		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 <= (-3.2d-88)) then
        tmp = -c / b
    else if (b <= 2.8d-120) 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 <= -3.2e-88) {
		tmp = -c / b;
	} else if (b <= 2.8e-120) {
		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 <= -3.2e-88:
		tmp = -c / b
	elif b <= 2.8e-120:
		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 <= -3.2e-88)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.8e-120)
		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 <= -3.2e-88)
		tmp = -c / b;
	elseif (b <= 2.8e-120)
		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, -3.2e-88], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.8e-120], 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 < -3.20000000000000012e-88

   1. Initial program 22.9%

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

   2. Taylor expanded in b around -inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   4. Simplified 86.9%

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

   if -3.20000000000000012e-88 < b < 2.79999999999999994e-120

   1. Initial program 79.4%

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

      1. frac-2neg 79.4%

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

      2. div-inv 79.3%

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

      3. *-commutative 79.3%

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

      4. distribute-lft-neg-in 79.3%

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

      5. associate-/r* 79.3%

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

      6. metadata-eval 79.3%

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

      7. metadata-eval 79.3%

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

      8. sub-neg 79.3%

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

      9. distribute-neg-out 79.3%

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

      10. remove-double-neg 79.3%

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

      11. fma-neg 79.3%

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

      12. distribute-lft-neg-in 79.3%

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

      13. *-commutative 79.3%

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

      14. associate-*l* 79.3%

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

      15. metadata-eval 79.3%

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

   3. Applied egg-rr 79.3%

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

   4. Taylor expanded in b around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*r* 74.5%

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

   6. Simplified 74.5%

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

   if 2.79999999999999994e-120 < b

   1. Initial program 67.2%

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

   2. Taylor expanded in b around inf 88.3%

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

      1. +-commutative 88.3%

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

      2. mul-1-neg 88.3%

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

      3. unsub-neg 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 67.5% accurate, 12.8× 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(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 <= -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 35.8%

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

   2. Taylor expanded in b around -inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

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

   4. Simplified 69.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.0%

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

   2. Taylor expanded in b around inf 78.3%

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

      1. +-commutative 78.3%

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

      2. mul-1-neg 78.3%

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

      3. unsub-neg 78.3%

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

   4. Simplified 78.3%

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

4. Final simplification 73.4%

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

Alternative 7: 42.7% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1.4e-20) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-20) {
		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 <= (-1.4d-20)) 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 <= -1.4e-20) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e-20:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e-20)
		tmp = 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 <= -1.4e-20)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e-20], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.4000000000000001e-20

   1. Initial program 18.6%

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

      1. frac-2neg 18.6%

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

      2. div-inv 18.6%

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

      3. *-commutative 18.6%

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

      4. distribute-lft-neg-in 18.6%

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

      5. associate-/r* 18.6%

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

      6. metadata-eval 18.6%

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

      7. metadata-eval 18.6%

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

      8. sub-neg 18.6%

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

      9. distribute-neg-out 18.6%

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

      10. remove-double-neg 18.6%

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

      11. fma-neg 18.6%

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

      12. distribute-lft-neg-in 18.6%

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

      13. *-commutative 18.6%

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

      14. associate-*l* 18.6%

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

      15. metadata-eval 18.6%

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

   3. Applied egg-rr 18.6%

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

   4. Taylor expanded in b around inf 2.4%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 2.4%

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

      2. *-commutative 2.4%

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

      3. associate-*l* 2.4%

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

      4. *-commutative 2.4%

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

   6. Simplified 2.4%

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

   7. Taylor expanded in a around inf 34.6%

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

   if -1.4000000000000001e-20 < b

   1. Initial program 68.9%

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

   2. Taylor expanded in b around inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. neg-mul-1 55.2%

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

   4. Simplified 55.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 67.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-301) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-301) {
		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 <= (-2.6d-301)) 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 <= -2.6e-301) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-301:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-301)
		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 <= -2.6e-301)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-301], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.5999999999999998e-301

   1. Initial program 35.3%

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

   2. Taylor expanded in b around -inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

   4. Simplified 69.8%

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

   if -2.5999999999999998e-301 < b

   1. Initial program 70.2%

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

   2. Taylor expanded in b around inf 76.8%

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

      1. associate-*r/ 76.8%

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

      2. neg-mul-1 76.8%

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

   4. Simplified 76.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9: 13.1% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -4.0) (/ c b) (/ -0.5 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.0) {
		tmp = c / b;
	} else {
		tmp = -0.5 / 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 <= (-4.0d0)) then
        tmp = c / b
    else
        tmp = (-0.5d0) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.0) {
		tmp = c / b;
	} else {
		tmp = -0.5 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.0:
		tmp = c / b
	else:
		tmp = -0.5 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.0)
		tmp = Float64(c / b);
	else
		tmp = Float64(-0.5 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.0)
		tmp = c / b;
	else
		tmp = -0.5 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.0], N[(c / b), $MachinePrecision], N[(-0.5 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4

   1. Initial program 18.2%

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

      1. frac-2neg 18.2%

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

      2. div-inv 18.2%

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

      3. *-commutative 18.2%

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

      4. distribute-lft-neg-in 18.2%

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

      5. associate-/r* 18.2%

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

      6. metadata-eval 18.2%

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

      7. metadata-eval 18.2%

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

      8. sub-neg 18.2%

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

      9. distribute-neg-out 18.2%

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

      10. remove-double-neg 18.2%

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

      11. fma-neg 18.2%

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

      12. distribute-lft-neg-in 18.2%

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

      13. *-commutative 18.2%

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

      14. associate-*l* 18.2%

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

      15. metadata-eval 18.2%

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

   3. Applied egg-rr 18.2%

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

   4. Taylor expanded in b around inf 2.4%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r/ 2.5%

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

      2. *-commutative 2.5%

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

      3. associate-*l* 2.5%

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

      4. *-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in a around inf 36.1%

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

   if -4 < b

   1. Initial program 67.9%

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

   3. Applied egg-rr 56.1%

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

      1. *-lft-identity 56.1%

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

      2. *-commutative 56.1%

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

   5. Applied egg-rr 61.0%

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

   7. Simplified 6.0%

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

4. Final simplification 16.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a}\\ \end{array} \]

Alternative 10: 5.4% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ -0.5 a))

C

double code(double a, double b, double c) {
	return -0.5 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) / a
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 / a;
}

Python

def code(a, b, c):
	return -0.5 / a

Julia

function code(a, b, c)
	return Float64(-0.5 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 / a;
end

Wolfram

code[a_, b_, c_] := N[(-0.5 / a), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

3. Applied egg-rr 43.2%

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

   1. *-lft-identity 43.2%

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

   2. *-commutative 43.2%

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

5. Applied egg-rr 50.9%

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

7. Simplified 5.2%

   \[\leadsto \color{blue}{\frac{-0.5}{a}} \]

8. Final simplification 5.2%

   \[\leadsto \frac{-0.5}{a} \]

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 2023297 
(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)))