Quadratic roots, narrow range

Percentage Accurate: 55.5% → 91.8%

Time: 11.7s

Alternatives: 10

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: 55.5% 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: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right) - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.35)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (-
      (-
       (fma
        -2.0
        (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
        (* -0.25 (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))))
       (/ c b))
      (* a (/ c (/ (pow b 3.0) c)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.35) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (fma(-2.0, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), (-0.25 * (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))))) - (c / b)) - (a * (c / (pow(b, 3.0) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.35)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-2.0, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), Float64(-0.25 * Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))))) - Float64(c / b)) - Float64(a * Float64(c / Float64((b ^ 3.0) / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.35], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $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 4 a) c)))) (*.f64 2 a)) < -0.34999999999999998

   1. Initial program 85.6%

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

      1. flip-+ 85.5%

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

      2. pow2 85.5%

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

      3. add-sqr-sqrt 87.4%

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

      4. *-commutative 87.4%

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

      5. *-commutative 87.4%

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

      6. *-commutative 87.4%

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

      7. *-commutative 87.4%

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

   3. Applied egg-rr 87.4%

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

   if -0.34999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 50.1%

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

      1. *-commutative 50.1%

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

      2. +-commutative 50.1%

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

      3. unsub-neg 50.1%

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

      4. fma-neg 50.2%

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

      5. associate-*l* 50.2%

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

      6. *-commutative 50.2%

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

      7. distribute-rgt-neg-in 50.2%

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

      8. metadata-eval 50.2%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in a around 0 92.4%

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

   6. Simplified 92.4%

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

   7. Taylor expanded in b around 0 92.4%

      \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \color{blue}{\frac{{a}^{3} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   8. Step-by-step derivation

      1. associate-/l* 92.4%

         \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{4 \cdot {c}^{4} + 16 \cdot {c}^{4}}}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

      2. distribute-rgt-out 92.4%

         \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(4 + 16\right)}}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

      3. metadata-eval 92.4%

         \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{20}}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

   9. Simplified 92.4%

      \[\leadsto \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right) - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 2: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(-2 \cdot \left(a \cdot a\right)\right) - \frac{c}{b}\right) - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.35)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (-
      (- (* (/ (pow c 3.0) (pow b 5.0)) (* -2.0 (* a a))) (/ c b))
      (* (* a c) (/ c (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.35) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (((pow(c, 3.0) / pow(b, 5.0)) * (-2.0 * (a * a))) - (c / b)) - ((a * c) * (c / pow(b, 3.0)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = sqrt(((b * b) - t_0))
    if (((t_1 - b) / (a * 2.0d0)) <= (-0.35d0)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = ((((c ** 3.0d0) / (b ** 5.0d0)) * ((-2.0d0) * (a * a))) - (c / b)) - ((a * c) * (c / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.35) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (((Math.pow(c, 3.0) / Math.pow(b, 5.0)) * (-2.0 * (a * a))) - (c / b)) - ((a * c) * (c / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -0.35:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = (((math.pow(c, 3.0) / math.pow(b, 5.0)) * (-2.0 * (a * a))) - (c / b)) - ((a * c) * (c / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.35)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(-2.0 * Float64(a * a))) - Float64(c / b)) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -0.35)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	else
		tmp = ((((c ^ 3.0) / (b ^ 5.0)) * (-2.0 * (a * a))) - (c / b)) - ((a * c) * (c / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.35], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $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 4 a) c)))) (*.f64 2 a)) < -0.34999999999999998

   1. Initial program 85.6%

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

      1. flip-+ 85.5%

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

      2. pow2 85.5%

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

      3. add-sqr-sqrt 87.4%

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

      4. *-commutative 87.4%

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

      5. *-commutative 87.4%

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

      6. *-commutative 87.4%

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

      7. *-commutative 87.4%

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

   3. Applied egg-rr 87.4%

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

   if -0.34999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 50.1%

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

      1. add-log-exp 47.9%

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

      2. neg-mul-1 47.9%

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

      3. fma-def 47.9%

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

      4. *-commutative 47.9%

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

      5. *-commutative 47.9%

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

      6. *-commutative 47.9%

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

   3. Applied egg-rr 47.9%

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

   4. Taylor expanded in b around inf 90.2%

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

      1. +-commutative 90.2%

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

      2. mul-1-neg 90.2%

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

      3. unsub-neg 90.2%

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

      4. +-commutative 90.2%

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

      5. mul-1-neg 90.2%

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

      6. unsub-neg 90.2%

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

      7. *-commutative 90.2%

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

      8. unpow2 90.2%

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

      9. associate-*l/ 90.2%

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

      10. associate-*l* 90.2%

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

      11. *-commutative 90.2%

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

      12. associate-*r/ 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(-2 \cdot \left(a \cdot a\right)\right) - \frac{c}{b}\right) - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \]

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -3.6e-5)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (- (/ (- c) b) (* (* a c) (/ c (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -3.6e-5) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * c) * (c / pow(b, 3.0)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = sqrt(((b * b) - t_0))
    if (((t_1 - b) / (a * 2.0d0)) <= (-3.6d-5)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a * c) * (c / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -3.6e-5) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * c) * (c / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -3.6e-5:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a * c) * (c / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -3.6e-5)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (4.0 * a);
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -3.6e-5)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	else
		tmp = (-c / b) - ((a * c) * (c / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -3.6e-5], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $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 4 a) c)))) (*.f64 2 a)) < -3.60000000000000009e-5

   1. Initial program 77.2%

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

      1. flip-+ 77.3%

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

      2. pow2 77.3%

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

      3. add-sqr-sqrt 78.9%

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

      4. *-commutative 78.9%

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

      5. *-commutative 78.9%

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

      6. *-commutative 78.9%

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

      7. *-commutative 78.9%

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

   3. Applied egg-rr 78.9%

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

   if -3.60000000000000009e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 38.6%

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

      1. neg-sub0 38.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 38.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 38.6%

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

      4. neg-mul-1 38.6%

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

      5. associate-*l/ 38.6%

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

      6. *-commutative 38.6%

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

      7. associate-/r* 38.6%

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

      8. /-rgt-identity 38.6%

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

      9. metadata-eval 38.6%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in b around inf 93.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + 2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{-0.5}{a} \]
   5. Step-by-step derivation

      1. distribute-lft-out 93.0%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)} \cdot \frac{-0.5}{a} \]

      2. associate-/l* 93.0%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      3. associate-/r/ 93.0%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      4. associate-/l* 93.0%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right) \cdot \frac{-0.5}{a} \]

      5. associate-/r/ 93.0%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]

      6. unpow2 93.0%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      7. associate-/l* 93.0%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      8. unpow2 93.0%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \frac{-0.5}{a} \]

   6. Simplified 93.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)} \cdot \frac{-0.5}{a} \]

   7. Taylor expanded in c around 0 93.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   8. Step-by-step derivation

      1. +-commutative 93.2%

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

      2. mul-1-neg 93.2%

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

      3. unsub-neg 93.2%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 93.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 93.2%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. *-commutative 93.2%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      7. associate-*r/ 93.2%

         \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

      8. unpow2 93.2%

         \[\leadsto \frac{-c}{b} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} \]

      9. associate-*l/ 93.2%

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

      10. *-commutative 93.2%

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

      11. associate-*r* 93.2%

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

   9. Simplified 93.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(c \cdot \left(4 \cdot a\right) - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \]

Alternative 4: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0003:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.0003)
   (* (- (sqrt (fma b b (* (* a c) -4.0))) b) (/ 0.5 a))
   (- (/ (- c) b) (* (* a c) (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.0003) {
		tmp = (sqrt(fma(b, b, ((a * c) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((a * c) * (c / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.0003)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.0003], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $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 4 a) c)))) (*.f64 2 a)) < -2.99999999999999974e-4

   1. Initial program 78.3%

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

      1. /-rgt-identity 78.3%

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

      2. metadata-eval 78.3%

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

      3. associate-/l* 78.3%

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

      4. associate-*r/ 78.3%

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

      5. +-commutative 78.3%

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

      6. unsub-neg 78.3%

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

      7. fma-neg 78.4%

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

      8. associate-*l* 78.4%

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

      9. *-commutative 78.4%

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

      10. distribute-rgt-neg-in 78.4%

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

      11. metadata-eval 78.4%

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

      12. associate-/r* 78.4%

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

      13. metadata-eval 78.4%

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

      14. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 41.8%

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

      1. neg-sub0 41.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 41.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 41.8%

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

      4. neg-mul-1 41.8%

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

      5. associate-*l/ 41.8%

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

      6. *-commutative 41.8%

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

      7. associate-/r* 41.8%

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

      8. /-rgt-identity 41.8%

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

      9. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in b around inf 90.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + 2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{-0.5}{a} \]
   5. Step-by-step derivation

      1. distribute-lft-out 90.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)} \cdot \frac{-0.5}{a} \]

      2. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      3. associate-/r/ 90.7%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      4. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right) \cdot \frac{-0.5}{a} \]

      5. associate-/r/ 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]

      6. unpow2 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      7. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      8. unpow2 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \frac{-0.5}{a} \]

   6. Simplified 90.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)} \cdot \frac{-0.5}{a} \]

   7. Taylor expanded in c around 0 90.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   8. Step-by-step derivation

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 90.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 90.9%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. *-commutative 90.9%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      7. associate-*r/ 90.9%

         \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

      8. unpow2 90.9%

         \[\leadsto \frac{-c}{b} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} \]

      9. associate-*l/ 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0003:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \]

Alternative 5: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0003:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.0003)
   (/ (- (sqrt (fma b b (* (* a c) -4.0))) b) (* a 2.0))
   (- (/ (- c) b) (* (* a c) (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.0003) {
		tmp = (sqrt(fma(b, b, ((a * c) * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * c) * (c / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.0003)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.0003], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $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 4 a) c)))) (*.f64 2 a)) < -2.99999999999999974e-4

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. +-commutative 78.3%

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

      3. unsub-neg 78.3%

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

      4. fma-neg 78.4%

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

      5. associate-*l* 78.4%

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

      6. *-commutative 78.4%

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

      7. distribute-rgt-neg-in 78.4%

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

      8. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 41.8%

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

      1. neg-sub0 41.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 41.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 41.8%

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

      4. neg-mul-1 41.8%

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

      5. associate-*l/ 41.8%

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

      6. *-commutative 41.8%

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

      7. associate-/r* 41.8%

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

      8. /-rgt-identity 41.8%

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

      9. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in b around inf 90.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + 2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{-0.5}{a} \]
   5. Step-by-step derivation

      1. distribute-lft-out 90.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)} \cdot \frac{-0.5}{a} \]

      2. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      3. associate-/r/ 90.7%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      4. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right) \cdot \frac{-0.5}{a} \]

      5. associate-/r/ 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]

      6. unpow2 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      7. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      8. unpow2 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \frac{-0.5}{a} \]

   6. Simplified 90.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)} \cdot \frac{-0.5}{a} \]

   7. Taylor expanded in c around 0 90.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   8. Step-by-step derivation

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 90.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 90.9%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. *-commutative 90.9%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      7. associate-*r/ 90.9%

         \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

      8. unpow2 90.9%

         \[\leadsto \frac{-c}{b} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} \]

      9. associate-*l/ 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0003:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \]

Alternative 6: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0003:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.0003)
   (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (* a c) (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.0003) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * c) * (c / pow(b, 3.0)));
	}
	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 (((sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)) <= (-0.0003d0)) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a * c) * (c / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.0003) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * c) * (c / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.0003:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a * c) * (c / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.0003)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.0003)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a * c) * (c / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.0003], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $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 4 a) c)))) (*.f64 2 a)) < -2.99999999999999974e-4

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. +-commutative 78.3%

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

      3. unsub-neg 78.3%

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

      4. fma-neg 78.4%

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

      5. associate-*l* 78.4%

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

      6. *-commutative 78.4%

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

      7. distribute-rgt-neg-in 78.4%

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

      8. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

      1. fma-udef 78.3%

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

      2. associate-*l* 78.3%

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

   5. Applied egg-rr 78.3%

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

   if -2.99999999999999974e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 41.8%

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

      1. neg-sub0 41.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 41.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 41.8%

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

      4. neg-mul-1 41.8%

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

      5. associate-*l/ 41.8%

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

      6. *-commutative 41.8%

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

      7. associate-/r* 41.8%

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

      8. /-rgt-identity 41.8%

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

      9. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in b around inf 90.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + 2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{-0.5}{a} \]
   5. Step-by-step derivation

      1. distribute-lft-out 90.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)} \cdot \frac{-0.5}{a} \]

      2. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      3. associate-/r/ 90.7%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      4. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right) \cdot \frac{-0.5}{a} \]

      5. associate-/r/ 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]

      6. unpow2 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      7. associate-/l* 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      8. unpow2 90.7%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \frac{-0.5}{a} \]

   6. Simplified 90.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)} \cdot \frac{-0.5}{a} \]

   7. Taylor expanded in c around 0 90.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   8. Step-by-step derivation

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 90.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 90.9%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. *-commutative 90.9%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      7. associate-*r/ 90.9%

         \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

      8. unpow2 90.9%

         \[\leadsto \frac{-c}{b} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} \]

      9. associate-*l/ 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-*r* 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0003:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \]

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.43:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.43)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))
   (- (/ (- c) b) (* (* a c) (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.43) {
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = (-c / b) - ((a * c) * (c / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.43d0) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b)
    else
        tmp = (-c / b) - ((a * c) * (c / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.43) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = (-c / b) - ((a * c) * (c / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.43:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + (a * (c * -4.0)))) - b)
	else:
		tmp = (-c / b) - ((a * c) * (c / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.43)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.43)
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	else
		tmp = (-c / b) - ((a * c) * (c / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.43], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.42999999999999994

   1. Initial program 84.5%

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

      1. /-rgt-identity 84.5%

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

      2. metadata-eval 84.5%

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

      3. associate-/l* 84.5%

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

      4. associate-*r/ 84.5%

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

      5. +-commutative 84.5%

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

      6. unsub-neg 84.5%

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

      7. fma-neg 84.6%

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

      8. associate-*l* 84.6%

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

      9. *-commutative 84.6%

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

      10. distribute-rgt-neg-in 84.6%

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

      11. metadata-eval 84.6%

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

      12. associate-/r* 84.6%

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

      13. metadata-eval 84.6%

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

      14. metadata-eval 84.6%

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

   3. Simplified 84.6%

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

      1. fma-udef 84.5%

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

      2. associate-*l* 84.5%

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

   5. Applied egg-rr 84.5%

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

   if 1.42999999999999994 < b

   1. Initial program 49.9%

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

      1. neg-sub0 49.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 49.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 49.9%

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

      4. neg-mul-1 49.9%

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

      5. associate-*l/ 49.9%

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

      6. *-commutative 49.9%

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

      7. associate-/r* 49.9%

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

      8. /-rgt-identity 49.9%

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

      9. metadata-eval 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in b around inf 84.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + 2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{-0.5}{a} \]
   5. Step-by-step derivation

      1. distribute-lft-out 84.3%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)} \cdot \frac{-0.5}{a} \]

      2. associate-/l* 84.3%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      3. associate-/r/ 84.3%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

      4. associate-/l* 84.3%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right) \cdot \frac{-0.5}{a} \]

      5. associate-/r/ 84.3%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]

      6. unpow2 84.3%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      7. associate-/l* 84.3%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

      8. unpow2 84.3%

         \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \frac{-0.5}{a} \]

   6. Simplified 84.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)} \cdot \frac{-0.5}{a} \]

   7. Taylor expanded in c around 0 84.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   8. Step-by-step derivation

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 84.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 84.5%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. *-commutative 84.5%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      7. associate-*r/ 84.5%

         \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

      8. unpow2 84.5%

         \[\leadsto \frac{-c}{b} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} \]

      9. associate-*l/ 84.5%

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

      10. *-commutative 84.5%

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

      11. associate-*r* 84.5%

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

   9. Simplified 84.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.43:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\\ \end{array} \]

Alternative 8: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} - \left(a \cdot c\right) \cdot \frac{c}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (/ (- c) b) (* (* a c) (/ c (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-c / b) - ((a * c) * (c / pow(b, 3.0)));
}

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) - ((a * c) * (c / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-c / b) - ((a * c) * (c / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-c / b) - ((a * c) * (c / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) / b) - Float64(Float64(a * c) * Float64(c / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-c / b) - ((a * c) * (c / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

   1. neg-sub0 56.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 56.2%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 56.2%

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

   4. neg-mul-1 56.2%

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

   5. associate-*l/ 56.2%

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

   6. *-commutative 56.2%

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

   7. associate-/r* 56.2%

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

   8. /-rgt-identity 56.2%

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

   9. metadata-eval 56.2%

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

3. Simplified 56.2%

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

4. Taylor expanded in b around inf 79.1%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + 2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{-0.5}{a} \]
5. Step-by-step derivation

   1. distribute-lft-out 79.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)} \cdot \frac{-0.5}{a} \]

   2. associate-/l* 79.1%

      \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

   3. associate-/r/ 79.1%

      \[\leadsto \left(2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right) \cdot \frac{-0.5}{a} \]

   4. associate-/l* 79.1%

      \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right) \cdot \frac{-0.5}{a} \]

   5. associate-/r/ 79.1%

      \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]

   6. unpow2 79.1%

      \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

   7. associate-/l* 79.1%

      \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right) \cdot \frac{-0.5}{a} \]

   8. unpow2 79.1%

      \[\leadsto \left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \frac{-0.5}{a} \]

6. Simplified 79.1%

   \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)} \cdot \frac{-0.5}{a} \]

7. Taylor expanded in c around 0 79.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
8. Step-by-step derivation

   1. +-commutative 79.3%

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

   2. mul-1-neg 79.3%

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

   3. unsub-neg 79.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   4. associate-*r/ 79.3%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   5. neg-mul-1 79.3%

      \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   6. *-commutative 79.3%

      \[\leadsto \frac{-c}{b} - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

   7. associate-*r/ 79.3%

      \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   8. unpow2 79.3%

      \[\leadsto \frac{-c}{b} - a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} \]

   9. associate-*l/ 79.3%

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

   10. *-commutative 79.3%

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

   11. associate-*r* 79.3%

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

9. Simplified 79.3%

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

10. Final simplification 79.3%

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

Alternative 9: 64.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. *-commutative 56.2%

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

   2. +-commutative 56.2%

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

   3. unsub-neg 56.2%

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

   4. fma-neg 56.3%

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

   5. associate-*l* 56.3%

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

   6. *-commutative 56.3%

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

   7. distribute-rgt-neg-in 56.3%

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

   8. metadata-eval 56.3%

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

3. Simplified 56.3%

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

4. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

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

   2. neg-mul-1 63.8%

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

6. Simplified 63.8%

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

7. Final simplification 63.8%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

   1. add-log-exp 50.1%

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

   2. neg-mul-1 50.1%

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

   3. fma-def 50.1%

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

   4. *-commutative 50.1%

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

   5. *-commutative 50.1%

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

   6. *-commutative 50.1%

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

3. Applied egg-rr 50.1%

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

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

6. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

7. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023240 
(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)))