Quadratic roots, medium range

Percentage Accurate: 31.7% → 95.4%

Time: 10.5s

Alternatives: 9

Speedup: 3.6×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.7% 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: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -5, {c}^{4}, \left(\left(\left(c \cdot a\right) \cdot -2 - b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma
    (* (* a a) -5.0)
    (pow c 4.0)
    (* (* (- (* (* c a) -2.0) (* b b)) (* c c)) (* b b)))
   (pow b 7.0))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma((fma(((a * a) * -5.0), pow(c, 4.0), (((((c * a) * -2.0) - (b * b)) * (c * c)) * (b * b))) / pow(b, 7.0)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(Float64(a * a) * -5.0), (c ^ 4.0), Float64(Float64(Float64(Float64(Float64(c * a) * -2.0) - Float64(b * b)) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -5.0), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -2.0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 94.9%

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

9. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
10. Step-by-step derivation

11. Applied rewrites 94.9%

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

12. Taylor expanded in c around 0

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \left({c}^{2} \cdot \left(-2 \cdot \left(a \cdot c\right) - {b}^{2}\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
13. Step-by-step derivation

14. Applied rewrites 94.9%

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

15. Final simplification 94.9%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -5, {c}^{4}, \left(\left(\left(c \cdot a\right) \cdot -2 - b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
16. Add Preprocessing

Alternative 2: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0} + b\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -160:\\ \;\;\;\;\frac{0.5}{\frac{a}{\frac{t\_0}{t\_1} - \frac{b \cdot b}{t\_1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (+ (sqrt t_0) b)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a)) -160.0)
     (/ 0.5 (/ a (- (/ t_0 t_1) (/ (* b b) t_1))))
     (/ 0.5 (/ (fma (* (/ c b) a) 0.5 (* -0.5 b)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0) + b;
	double tmp;
	if (((sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a)) <= -160.0) {
		tmp = 0.5 / (a / ((t_0 / t_1) - ((b * b) / t_1)));
	} else {
		tmp = 0.5 / (fma(((c / b) * a), 0.5, (-0.5 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = Float64(sqrt(t_0) + b)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(2.0 * a)) <= -160.0)
		tmp = Float64(0.5 / Float64(a / Float64(Float64(t_0 / t_1) - Float64(Float64(b * b) / t_1))));
	else
		tmp = Float64(0.5 / Float64(fma(Float64(Float64(c / b) * a), 0.5, Float64(-0.5 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -160.0], N[(0.5 / N[(a / N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 0.5 + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $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)) < -160

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.2

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

   4. Applied rewrites 81.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. sub-div N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   6. Applied rewrites 82.9%

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

   if -160 < (/.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 29.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 29.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 29.5

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -160:\\ \;\;\;\;\frac{0.5}{\frac{a}{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b} - \frac{b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, \frac{a}{b} \cdot 0.5\right), c, -0.5 \cdot b\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (/
   (fma (fma (/ (* 0.5 (* a a)) (pow b 3.0)) c (* (/ a b) 0.5)) c (* -0.5 b))
   c)))

C

double code(double a, double b, double c) {
	return 0.5 / (fma(fma(((0.5 * (a * a)) / pow(b, 3.0)), c, ((a / b) * 0.5)), c, (-0.5 * b)) / c);
}

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(fma(fma(Float64(Float64(0.5 * Float64(a * a)) / (b ^ 3.0)), c, Float64(Float64(a / b) * 0.5)), c, Float64(-0.5 * b)) / c))
end

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.4

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

4. Applied rewrites 33.4%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 93.1%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, \frac{a}{b} \cdot 0.5\right), c, -0.5 \cdot b\right)}{c}}} \]
8. Add Preprocessing

Alternative 4: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(\frac{b}{c}, -0.5, \mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.5, a, \frac{0.5}{b}\right) \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.5 (fma (/ b c) -0.5 (* (fma (* (/ c (pow b 3.0)) 0.5) a (/ 0.5 b)) a))))

C

double code(double a, double b, double c) {
	return 0.5 / fma((b / c), -0.5, (fma(((c / pow(b, 3.0)) * 0.5), a, (0.5 / b)) * a));
}

Julia

function code(a, b, c)
	return Float64(0.5 / fma(Float64(b / c), -0.5, Float64(fma(Float64(Float64(c / (b ^ 3.0)) * 0.5), a, Float64(0.5 / b)) * a)))
end

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.4

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

4. Applied rewrites 33.4%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 93.1%

   \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\frac{b}{c}, -0.5, \mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.5, a, \frac{0.5}{b}\right) \cdot a\right)}} \]
8. Add Preprocessing

Alternative 5: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -160:\\ \;\;\;\;\frac{0.5}{\frac{a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a)) -160.0)
     (/ 0.5 (/ a (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (/ 0.5 (/ (fma (* (/ c b) a) 0.5 (* -0.5 b)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a)) <= -160.0) {
		tmp = 0.5 / (a / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = 0.5 / (fma(((c / b) * a), 0.5, (-0.5 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(2.0 * a)) <= -160.0)
		tmp = Float64(0.5 / Float64(a / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(0.5 / Float64(fma(Float64(Float64(c / b) * a), 0.5, Float64(-0.5 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -160.0], N[(0.5 / N[(a / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 0.5 + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $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)) < -160

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.2

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

   4. Applied rewrites 81.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

   6. Applied rewrites 82.8%

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

   if -160 < (/.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 29.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 29.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 29.5

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -160:\\ \;\;\;\;\frac{0.5}{\frac{a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -160:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot 0.5}{\left(\sqrt{t\_0} + b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a)) -160.0)
     (/ (* (- t_0 (* b b)) 0.5) (* (+ (sqrt t_0) b) a))
     (/ 0.5 (/ (fma (* (/ c b) a) 0.5 (* -0.5 b)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a)) <= -160.0) {
		tmp = ((t_0 - (b * b)) * 0.5) / ((sqrt(t_0) + b) * a);
	} else {
		tmp = 0.5 / (fma(((c / b) * a), 0.5, (-0.5 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(2.0 * a)) <= -160.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * 0.5) / Float64(Float64(sqrt(t_0) + b) * a));
	else
		tmp = Float64(0.5 / Float64(fma(Float64(Float64(c / b) * a), 0.5, Float64(-0.5 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -160.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 0.5 + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $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)) < -160

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.2

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

   4. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 82.7%

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

   if -160 < (/.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 29.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 29.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 29.5

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a} \leq -160:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b\right) \cdot 0.5}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.4

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

4. Applied rewrites 33.4%

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

5. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 89.3

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

7. Applied rewrites 89.3%

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

8. Final simplification 89.3%

   \[\leadsto \frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}} \]
9. Add Preprocessing

Alternative 8: 90.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(\frac{b}{c}, -0.5, \frac{a}{b} \cdot 0.5\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 33.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 33.4

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

4. Applied rewrites 33.4%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 89.3

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

7. Applied rewrites 89.3%

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

Alternative 9: 81.2% accurate, 3.6× 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 33.4%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   4. lower-neg.f64 79.2

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

5. Applied rewrites 79.2%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))