Quadratic roots, full range

Percentage Accurate: 52.1% → 84.9%

Time: 15.6s

Alternatives: 6

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+147}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+147)
   (- (/ b a))
   (if (<= b 5.6e-34)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+147) {
		tmp = -(b / a);
	} else if (b <= 5.6e-34) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+147)) then
        tmp = -(b / a)
    else if (b <= 5.6d-34) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+147) {
		tmp = -(b / a);
	} else if (b <= 5.6e-34) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+147:
		tmp = -(b / a)
	elif b <= 5.6e-34:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+147)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 5.6e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+147)
		tmp = -(b / a);
	elseif (b <= 5.6e-34)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+147], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 5.6e-34], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000002e147

   1. Initial program 42.2%

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

      1. *-commutative 42.2%

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

   3. Simplified 42.2%

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

   5. Taylor expanded in b around -inf 94.0%

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

      1. associate-*r/ 94.0%

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

      2. mul-1-neg 94.0%

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

   7. Simplified 94.0%

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

   if -5.0000000000000002e147 < b < 5.59999999999999994e-34

   1. Initial program 81.9%

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

   if 5.59999999999999994e-34 < b

   1. Initial program 7.4%

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

      1. *-commutative 7.4%

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

   3. Simplified 7.4%

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

      1. clear-num 7.4%

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

      2. inv-pow 7.4%

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

   6. Applied egg-rr 2.9%

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

      1. unpow-1 2.9%

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

   8. Simplified 2.9%

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

      1. associate-/l* 2.9%

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

      2. frac-2neg 2.9%

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

      3. metadata-eval 2.9%

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

      4. hypot-undefine 2.7%

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

      5. add-sqr-sqrt 3.2%

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

      6. fma-undefine 3.2%

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

      7. add-cube-cbrt 3.2%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{-\left(b + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}\right)}} \]

      8. unpow3 3.2%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{-\left(b + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}\right)}} \]

      9. distribute-neg-in 3.2%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\left(-b\right) + \left(-\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}\right)}}} \]

      10. sub-neg 3.2%

          \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\left(-b\right) - \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}}} \]

      11. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}} \]

   10. Applied egg-rr 17.1%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. +-commutative 0.0%

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

       2. associate-*r/ 0.0%

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

       3. *-commutative 0.0%

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

       4. times-frac 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 94.2%

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

       7. metadata-eval 94.2%

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

       8. neg-mul-1 94.2%

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

       9. distribute-neg-frac2 94.2%

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

   13. Simplified 94.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} + \frac{b}{-c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+147}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-82)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 2.7e-30)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-82) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2.7e-30) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.5d-82)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 2.7d-30) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-82) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2.7e-30) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-82:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 2.7e-30:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-82)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 2.7e-30)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-82)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 2.7e-30)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-82], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e-30], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4999999999999999e-82

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. *-commutative 89.3%

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

      3. distribute-rgt-neg-in 89.3%

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

      4. +-commutative 89.3%

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

      5. mul-1-neg 89.3%

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

      6. unsub-neg 89.3%

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

   7. Simplified 89.3%

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

   if -2.4999999999999999e-82 < b < 2.69999999999999987e-30

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in b around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*r* 66.6%

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

   7. Simplified 66.6%

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

   if 2.69999999999999987e-30 < b

   1. Initial program 7.4%

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

      1. *-commutative 7.4%

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

   3. Simplified 7.4%

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

      1. clear-num 7.4%

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

      2. inv-pow 7.4%

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

   6. Applied egg-rr 2.9%

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

      1. unpow-1 2.9%

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

   8. Simplified 2.9%

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

      1. associate-/l* 2.9%

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

      2. frac-2neg 2.9%

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

      3. metadata-eval 2.9%

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

      4. hypot-undefine 2.7%

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

      5. add-sqr-sqrt 3.2%

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

      6. fma-undefine 3.2%

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

      7. add-cube-cbrt 3.2%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{-\left(b + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}\right)}} \]

      8. unpow3 3.2%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{-\left(b + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}\right)}} \]

      9. distribute-neg-in 3.2%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\left(-b\right) + \left(-\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}\right)}}} \]

      10. sub-neg 3.2%

          \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\left(-b\right) - \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}}} \]

      11. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}} \]

   10. Applied egg-rr 17.1%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. +-commutative 0.0%

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

       2. associate-*r/ 0.0%

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

       3. *-commutative 0.0%

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

       4. times-frac 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 94.2%

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

       7. metadata-eval 94.2%

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

       8. neg-mul-1 94.2%

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

       9. distribute-neg-frac2 94.2%

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

   13. Simplified 94.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} + \frac{b}{-c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-302}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-302) (- (/ b a)) (/ -1.0 (- (/ b c) (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-302) {
		tmp = -(b / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.6d-302)) then
        tmp = -(b / a)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-302) {
		tmp = -(b / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-302:
		tmp = -(b / a)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-302)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e-302)
		tmp = -(b / a);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-302], (-N[(b / a), $MachinePrecision]), N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.6000000000000001e-302

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in b around -inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. mul-1-neg 70.0%

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

   7. Simplified 70.0%

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

   if -3.6000000000000001e-302 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

      1. clear-num 26.9%

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

      2. inv-pow 26.9%

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

   6. Applied egg-rr 23.7%

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

      1. unpow-1 23.7%

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

   8. Simplified 23.7%

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

      1. associate-/l* 23.7%

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

      2. frac-2neg 23.7%

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

      3. metadata-eval 23.7%

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

      4. hypot-undefine 23.5%

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

      5. add-sqr-sqrt 23.9%

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

      6. fma-undefine 23.9%

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

      7. add-cube-cbrt 23.7%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{-\left(b + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}\right)}} \]

      8. unpow3 23.7%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{-\left(b + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}\right)}} \]

      9. distribute-neg-in 23.7%

         \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\left(-b\right) + \left(-\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}\right)}}} \]

      10. sub-neg 23.7%

          \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\left(-b\right) - \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}}} \]

      11. add-sqr-sqrt 0.8%

          \[\leadsto \frac{1}{a \cdot \frac{-2}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3}}}} \]

   10. Applied egg-rr 33.4%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. +-commutative 0.0%

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

       2. associate-*r/ 0.0%

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

       3. *-commutative 0.0%

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

       4. times-frac 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 73.0%

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

       7. metadata-eval 73.0%

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

       8. neg-mul-1 73.0%

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

       9. distribute-neg-frac2 73.0%

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

   13. Simplified 73.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} + \frac{b}{-c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-302}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in b around -inf 69.0%

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

      1. associate-*r/ 69.0%

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

      2. mul-1-neg 69.0%

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

   7. Simplified 69.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.5%

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

      1. *-commutative 26.5%

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

   3. Simplified 26.5%

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

   5. Taylor expanded in b around inf 73.8%

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

      1. associate-*r/ 73.8%

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

      2. mul-1-neg 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 71.2%

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

Alternative 5: 34.7% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

   1. *-commutative 50.6%

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

3. Simplified 50.6%

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

5. Taylor expanded in b around -inf 38.2%

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

   1. associate-*r/ 38.2%

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

   2. mul-1-neg 38.2%

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

7. Simplified 38.2%

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

8. Final simplification 38.2%

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

Alternative 6: 2.6% 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 50.6%

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

   1. *-commutative 50.6%

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

3. Simplified 50.6%

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

   1. clear-num 50.5%

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

   2. inv-pow 50.5%

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

6. Applied egg-rr 22.2%

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

   1. unpow-1 22.2%

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

8. Simplified 22.2%

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

9. Taylor expanded in a around 0 2.3%

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

10. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2024080 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))