Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.7%

Time: 15.7s

Alternatives: 9

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.4% 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: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -1.2)
     t_0
     (-
      (*
       (pow c 4.0)
       (-
        (* -5.0 (/ (pow a 3.0) (pow b 7.0)))
        (/ (+ (* 2.0 (/ (pow a 2.0) (pow b 5.0))) (/ a (* c (pow b 3.0)))) c)))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = (pow(c, 4.0) * ((-5.0 * (pow(a, 3.0) / pow(b, 7.0))) - (((2.0 * (pow(a, 2.0) / pow(b, 5.0))) + (a / (c * pow(b, 3.0)))) / c))) - (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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-1.2d0)) then
        tmp = t_0
    else
        tmp = ((c ** 4.0d0) * (((-5.0d0) * ((a ** 3.0d0) / (b ** 7.0d0))) - (((2.0d0 * ((a ** 2.0d0) / (b ** 5.0d0))) + (a / (c * (b ** 3.0d0)))) / c))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(c, 4.0) * ((-5.0 * (Math.pow(a, 3.0) / Math.pow(b, 7.0))) - (((2.0 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) + (a / (c * Math.pow(b, 3.0)))) / c))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -1.2:
		tmp = t_0
	else:
		tmp = (math.pow(c, 4.0) * ((-5.0 * (math.pow(a, 3.0) / math.pow(b, 7.0))) - (((2.0 * (math.pow(a, 2.0) / math.pow(b, 5.0))) + (a / (c * math.pow(b, 3.0)))) / c))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = Float64(Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64((a ^ 3.0) / (b ^ 7.0))) - Float64(Float64(Float64(2.0 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(a / Float64(c * (b ^ 3.0)))) / c))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = ((c ^ 4.0) * ((-5.0 * ((a ^ 3.0) / (b ^ 7.0))) - (((2.0 * ((a ^ 2.0) / (b ^ 5.0))) + (a / (c * (b ^ 3.0)))) / c))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2], t$95$0, N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 89.1%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in a around 0 93.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   6. Taylor expanded in c around -inf 93.8%

      \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + \frac{{a}^{2}}{{b}^{5}} \cdot -2\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -1.2)
     t_0
     (*
      c
      (+
       (*
        c
        (-
         (*
          c
          (+
           (* -5.0 (/ (* c (pow a 3.0)) (pow b 7.0)))
           (* (/ (pow a 2.0) (pow b 5.0)) -2.0)))
         (/ a (pow b 3.0))))
       (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = c * ((c * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 7.0))) + ((pow(a, 2.0) / pow(b, 5.0)) * -2.0))) - (a / pow(b, 3.0)))) + (-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-1.2d0)) then
        tmp = t_0
    else
        tmp = c * ((c * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 7.0d0))) + (((a ** 2.0d0) / (b ** 5.0d0)) * (-2.0d0)))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = c * ((c * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 7.0))) + ((Math.pow(a, 2.0) / Math.pow(b, 5.0)) * -2.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -1.2:
		tmp = t_0
	else:
		tmp = c * ((c * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 7.0))) + ((math.pow(a, 2.0) / math.pow(b, 5.0)) * -2.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))) + Float64(Float64((a ^ 2.0) / (b ^ 5.0)) * -2.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = c * ((c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 7.0))) + (((a ^ 2.0) / (b ^ 5.0)) * -2.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2], t$95$0, N[(c * N[(N[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 89.1%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in c around 0 93.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

   7. Simplified 93.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a \cdot b}\right)\right)\right) - \frac{1}{b}\right)} \]

   8. Taylor expanded in c around 0 93.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + \frac{{a}^{2}}{{b}^{5}} \cdot -2\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + \mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, \frac{2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{4}}\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -1.2)
     t_0
     (/
      (+
       c
       (fma
        a
        (pow (/ c b) 2.0)
        (/ (* 2.0 (* (pow a 2.0) (pow c 3.0))) (pow b 4.0))))
      (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = (c + fma(a, pow((c / b), 2.0), ((2.0 * (pow(a, 2.0) * pow(c, 3.0))) / pow(b, 4.0)))) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(c + fma(a, (Float64(c / b) ^ 2.0), Float64(Float64(2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 4.0)))) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2], t$95$0, N[(N[(c + N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 89.1%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in c around 0 90.2%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-4 \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + -2 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}}{a \cdot 2} \]

   6. Taylor expanded in b around -inf 90.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   7. Step-by-step derivation

      1. mul-1-neg 90.5%

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

      2. distribute-neg-frac2 90.5%

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

      3. +-commutative 90.5%

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

      4. associate-/l* 90.5%

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

      5. fma-define 90.5%

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

      6. unpow2 90.5%

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

      7. unpow2 90.5%

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

      8. times-frac 90.5%

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

      9. unpow2 90.5%

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

      10. associate-*r/ 90.5%

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

      11. *-commutative 90.5%

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

   8. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{c + \mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, \frac{2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{4}}\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + \mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, \frac{2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{4}}\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot -2\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -1.2)
     t_0
     (-
      (*
       a
       (-
        (* (/ (pow c 3.0) (pow b 5.0)) (* a -2.0))
        (/ (pow c 2.0) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = (a * (((pow(c, 3.0) / pow(b, 5.0)) * (a * -2.0)) - (pow(c, 2.0) / pow(b, 3.0)))) - (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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-1.2d0)) then
        tmp = t_0
    else
        tmp = (a * ((((c ** 3.0d0) / (b ** 5.0d0)) * (a * (-2.0d0))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = (a * (((Math.pow(c, 3.0) / Math.pow(b, 5.0)) * (a * -2.0)) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -1.2:
		tmp = t_0
	else:
		tmp = (a * (((math.pow(c, 3.0) / math.pow(b, 5.0)) * (a * -2.0)) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(a * Float64(Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * -2.0)) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = (a * ((((c ^ 3.0) / (b ^ 5.0)) * (a * -2.0)) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2], t$95$0, N[(N[(a * N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 89.1%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in c around 0 93.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

   7. Simplified 93.6%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a \cdot b}\right)\right)\right) - \frac{1}{b}\right)} \]

   8. Taylor expanded in a around 0 90.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 90.4%

         \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right) + -1 \cdot \frac{c}{b}} \]

      2. mul-1-neg 90.4%

         \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{\left(-\frac{c}{b}\right)} \]

      3. unsub-neg 90.4%

         \[\leadsto \color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

      4. mul-1-neg 90.4%

         \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)}\right) - \frac{c}{b} \]

      5. unsub-neg 90.4%

         \[\leadsto a \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

      6. associate-/l* 90.4%

         \[\leadsto a \cdot \left(-2 \cdot \color{blue}{\left(a \cdot \frac{{c}^{3}}{{b}^{5}}\right)} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

      7. *-commutative 90.4%

         \[\leadsto a \cdot \left(-2 \cdot \color{blue}{\left(\frac{{c}^{3}}{{b}^{5}} \cdot a\right)} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

      8. associate-*l* 90.4%

         \[\leadsto a \cdot \left(\color{blue}{\left(-2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) \cdot a} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

      9. *-commutative 90.4%

         \[\leadsto a \cdot \left(\color{blue}{\left(\frac{{c}^{3}}{{b}^{5}} \cdot -2\right)} \cdot a - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

      10. associate-*l* 90.4%

          \[\leadsto a \cdot \left(\color{blue}{\frac{{c}^{3}}{{b}^{5}} \cdot \left(-2 \cdot a\right)} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

   10. Simplified 90.4%

       \[\leadsto \color{blue}{a \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(-2 \cdot a\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot -2\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -1.2)
     t_0
     (*
      c
      (+
       (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
       (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-1.2d0)) then
        tmp = t_0
    else
        tmp = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.2) {
		tmp = t_0;
	} else {
		tmp = c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -1.2:
		tmp = t_0
	else:
		tmp = c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -1.2)
		tmp = t_0;
	else
		tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2], t$95$0, N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.19999999999999996

   1. Initial program 89.1%

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

   if -1.19999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in c around 0 90.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.05)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (* 2.0 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.05) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.05)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.050000000000000003

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. +-commutative 83.3%

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

      3. sqr-neg 83.3%

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

      4. unsub-neg 83.3%

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

      5. sqr-neg 83.3%

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

      6. fma-neg 83.4%

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

      7. distribute-lft-neg-in 83.4%

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

      8. *-commutative 83.4%

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

      9. *-commutative 83.4%

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

      10. distribute-rgt-neg-in 83.4%

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

      11. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

   if -0.050000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in a around 0 86.6%

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

      1. distribute-lft-out 86.6%

         \[\leadsto \frac{a \cdot \color{blue}{\left(-2 \cdot \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)}}{a \cdot 2} \]

      2. associate-/l* 86.6%

         \[\leadsto \frac{a \cdot \left(-2 \cdot \left(\frac{c}{b} + \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}\right)\right)}{a \cdot 2} \]

   7. Simplified 86.6%

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

      1. clear-num 86.5%

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

      2. inv-pow 86.5%

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

      3. *-commutative 86.5%

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

      4. associate-*r* 86.5%

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

      5. +-commutative 86.5%

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

      6. *-commutative 86.5%

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

      7. fma-define 86.5%

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

      8. div-inv 86.5%

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

      9. pow-flip 86.5%

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

      10. metadata-eval 86.5%

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

   9. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\left(a \cdot -2\right) \cdot \mathsf{fma}\left({c}^{2} \cdot {b}^{-3}, a, \frac{c}{b}\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 86.5%

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

       2. associate-/l* 86.5%

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

   11. Simplified 86.5%

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

   12. Taylor expanded in a around 0 87.2%

       \[\leadsto \frac{1}{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.05)
     t_0
     (/ 1.0 (* 2.0 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.05d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (2.0d0 * (((-0.5d0) * (b / c)) + (0.5d0 * (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.05:
		tmp = t_0
	else:
		tmp = 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = t_0;
	else
		tmp = 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], t$95$0, N[(1.0 / N[(2.0 * N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.050000000000000003

   1. Initial program 83.3%

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

   if -0.050000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in a around 0 86.6%

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

      1. distribute-lft-out 86.6%

         \[\leadsto \frac{a \cdot \color{blue}{\left(-2 \cdot \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)}}{a \cdot 2} \]

      2. associate-/l* 86.6%

         \[\leadsto \frac{a \cdot \left(-2 \cdot \left(\frac{c}{b} + \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}\right)\right)}{a \cdot 2} \]

   7. Simplified 86.6%

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

      1. clear-num 86.5%

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

      2. inv-pow 86.5%

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

      3. *-commutative 86.5%

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

      4. associate-*r* 86.5%

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

      5. +-commutative 86.5%

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

      6. *-commutative 86.5%

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

      7. fma-define 86.5%

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

      8. div-inv 86.5%

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

      9. pow-flip 86.5%

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

      10. metadata-eval 86.5%

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

   9. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\left(a \cdot -2\right) \cdot \mathsf{fma}\left({c}^{2} \cdot {b}^{-3}, a, \frac{c}{b}\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 86.5%

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

       2. associate-/l* 86.5%

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

   11. Simplified 86.5%

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

   12. Taylor expanded in a around 0 87.2%

       \[\leadsto \frac{1}{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.05:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 \cdot \left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (* 2.0 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))))

C

double code(double a, double b, double c) {
	return 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (2.0d0 * (((-0.5d0) * (b / c)) + (0.5d0 * (a / b))))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))));
}

Python

def code(a, b, c):
	return 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(2.0 * Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (2.0 * ((-0.5 * (b / c)) + (0.5 * (a / b))));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(2.0 * N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.3%

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

   1. *-commutative 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in a around 0 78.3%

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

   1. distribute-lft-out 78.3%

      \[\leadsto \frac{a \cdot \color{blue}{\left(-2 \cdot \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)}}{a \cdot 2} \]

   2. associate-/l* 78.3%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \left(\frac{c}{b} + \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}\right)\right)}{a \cdot 2} \]

7. Simplified 78.3%

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

   1. clear-num 78.2%

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

   2. inv-pow 78.2%

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

   3. *-commutative 78.2%

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

   4. associate-*r* 78.2%

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

   5. +-commutative 78.2%

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

   6. *-commutative 78.2%

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

   7. fma-define 78.2%

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

   8. div-inv 78.2%

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

   9. pow-flip 78.2%

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

   10. metadata-eval 78.2%

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

9. Applied egg-rr 78.2%

   \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\left(a \cdot -2\right) \cdot \mathsf{fma}\left({c}^{2} \cdot {b}^{-3}, a, \frac{c}{b}\right)}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 78.2%

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

    2. associate-/l* 78.2%

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

11. Simplified 78.2%

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

12. Taylor expanded in a around 0 79.1%

    \[\leadsto \frac{1}{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)}} \]

13. Final simplification 79.1%

    \[\leadsto \frac{1}{2 \cdot \left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \]
14. Add Preprocessing

Alternative 9: 64.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

   1. *-commutative 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in b around inf 61.4%

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

   1. associate-*r/ 61.4%

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

   2. mul-1-neg 61.4%

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

7. Simplified 61.4%

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

8. Final simplification 61.4%

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

Reproduce

herbie shell --seed 2024075 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))