Cubic critical, narrow range

Percentage Accurate: 55.7% → 99.1%

Time: 13.4s

Alternatives: 10

Speedup: 2.9×

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

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* -3.0 (* c a))
   (+ b (sqrt (* b (fma (* a (/ c (* b b))) (* -3.0 b) b)))))
  (* a 3.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in b around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unpow1 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. pow1/2 N/A

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

   12. lift-sqrt.f64 N/A

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

7. Applied rewrites 56.7%

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

8. Taylor expanded in b around 0

   \[\leadsto \frac{\frac{\color{blue}{-3 \cdot \left(a \cdot c\right)}}{\sqrt{b \cdot \mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, b \cdot -3, b\right)} - \left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

    \[\leadsto \frac{\frac{-3 \cdot \left(c \cdot a\right)}{b + \sqrt{b \cdot \mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -3 \cdot b, b\right)}}}{a \cdot 3} \]
12. Add Preprocessing

Alternative 2: 88.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 12.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (*
    c
    (fma
     c
     (/ (fma -0.5625 (/ (* a (* c a)) (* b b)) (* a -0.375)) (* b (* b b)))
     (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * fma(c, (fma(-0.5625, ((a * (c * a)) / (b * b)), (a * -0.375)) / (b * (b * b))), (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 12.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * fma(c, Float64(fma(-0.5625, Float64(Float64(a * Float64(c * a)) / Float64(b * b)), Float64(a * -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 12.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(c * N[(N[(-0.5625 * N[(N[(a * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 12

   1. Initial program 84.7%

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

   3. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 84.8

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

   8. Applied rewrites 84.8%

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

   if 12 < b

   1. Initial program 47.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)}{{b}^{5}}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{\color{blue}{\left(\frac{-9}{16} \cdot {a}^{2}\right) \cdot c}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{\frac{-9}{16} \cdot {a}^{2}}{{b}^{5}} \cdot c} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)} \cdot c + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot a, \frac{c}{{b}^{5}} \cdot -0.5625, \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto c \cdot \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{3}}}, \frac{\frac{-1}{2}}{b}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{3}}}, \frac{\frac{-1}{2}}{b}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\color{blue}{\mathsf{fma}\left(\frac{-9}{16}, \frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \color{blue}{\frac{{a}^{2} \cdot c}{{b}^{2}}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\color{blue}{\left(a \cdot a\right)} \cdot c}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      5. associate-*l* N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\color{blue}{a \cdot \left(a \cdot c\right)}}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\color{blue}{a \cdot \left(a \cdot c\right)}}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \color{blue}{\left(a \cdot c\right)}}{{b}^{2}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      8. unpow2 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{\color{blue}{b \cdot b}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{\color{blue}{b \cdot b}}, \frac{-3}{8} \cdot a\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      10. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, \color{blue}{a \cdot \frac{-3}{8}}\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, \color{blue}{a \cdot \frac{-3}{8}}\right)}{{b}^{3}}, \frac{\frac{-1}{2}}{b}\right) \]

      12. cube-mult N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, a \cdot \frac{-3}{8}\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{\frac{-1}{2}}{b}\right) \]

      13. unpow2 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, a \cdot \frac{-3}{8}\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{\frac{-1}{2}}{b}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, a \cdot \frac{-3}{8}\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{\frac{-1}{2}}{b}\right) \]

      15. unpow2 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, a \cdot \frac{-3}{8}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{\frac{-1}{2}}{b}\right) \]

      16. lower-*.f64 91.0

          \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-0.5}{b}\right) \]

   8. Applied rewrites 91.0%

      \[\leadsto c \cdot \mathsf{fma}\left(c, \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot c\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}}, \frac{-0.5}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1200.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (fma a (/ (* -0.375 (* c c)) (* b (* b b))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1200.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1200.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1200.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1200

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 1200 < b

   1. Initial program 40.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 91.8

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Applied rewrites 91.8%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1200.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/ (fma a (/ (* -0.375 (* c c)) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1200.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1200.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1200.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1200

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 1200 < b

   1. Initial program 40.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1200.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/ (* c (fma (* c a) (/ -0.375 (* b b)) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1200.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * fma((c * a), (-0.375 / (b * b)), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1200.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * fma(Float64(c * a), Float64(-0.375 / Float64(b * b)), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1200.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * a), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1200

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 1200 < b

   1. Initial program 40.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]

      2. sub-neg N/A

         \[\leadsto \frac{c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      4. *-commutative N/A

         \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{-3}{8}}}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      5. associate-/l* N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{c \cdot \left(\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}} + \color{blue}{\frac{-1}{2}}\right)}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{a \cdot c}, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \color{blue}{\frac{\frac{-3}{8}}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

      11. lower-*.f64 91.7

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

   8. Applied rewrites 91.7%

      \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{b \cdot b}, -0.5\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 115.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c (fma (* c a) (/ -0.375 (* b b)) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 115.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * fma((c * a), (-0.375 / (b * b)), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 115.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * fma(Float64(c * a), Float64(-0.375 / Float64(b * b)), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 115.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * a), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 115

   1. Initial program 81.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}{3 \cdot a} \]

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 81.9

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

   4. Applied rewrites 81.9%

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

   if 115 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]

      2. sub-neg N/A

         \[\leadsto \frac{c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      4. *-commutative N/A

         \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{-3}{8}}}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      5. associate-/l* N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{c \cdot \left(\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}} + \color{blue}{\frac{-1}{2}}\right)}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{a \cdot c}, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \color{blue}{\frac{\frac{-3}{8}}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

      11. lower-*.f64 88.4

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

   8. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{b \cdot b}, -0.5\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 115.0)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* a 3.0))
   (/ (* c (fma (* c a) (/ -0.375 (* b b)) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 115.0) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (a * 3.0);
	} else {
		tmp = (c * fma((c * a), (-0.375 / (b * b)), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 115.0)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * fma(Float64(c * a), Float64(-0.375 / Float64(b * b)), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 115.0], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * a), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 115

   1. Initial program 81.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 81.9

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

   4. Applied rewrites 81.9%

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

   if 115 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]

      2. sub-neg N/A

         \[\leadsto \frac{c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      4. *-commutative N/A

         \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{-3}{8}}}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      5. associate-/l* N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{c \cdot \left(\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}} + \color{blue}{\frac{-1}{2}}\right)}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{a \cdot c}, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \color{blue}{\frac{\frac{-3}{8}}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

      11. lower-*.f64 88.4

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

   8. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{b \cdot b}, -0.5\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 115.0)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (/ (* c (fma (* c a) (/ -0.375 (* b b)) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 115.0) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = (c * fma((c * a), (-0.375 / (b * b)), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 115.0)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * fma(Float64(c * a), Float64(-0.375 / Float64(b * b)), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 115.0], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * a), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 115

   1. Initial program 81.9%

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

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if 115 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]

      2. sub-neg N/A

         \[\leadsto \frac{c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      4. *-commutative N/A

         \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{-3}{8}}}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      5. associate-/l* N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b} \]

      6. metadata-eval N/A

         \[\leadsto \frac{c \cdot \left(\left(a \cdot c\right) \cdot \frac{\frac{-3}{8}}{{b}^{2}} + \color{blue}{\frac{-1}{2}}\right)}{b} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{a \cdot c}, \frac{\frac{-3}{8}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \color{blue}{\frac{\frac{-3}{8}}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{\frac{-3}{8}}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

      11. lower-*.f64 88.4

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

   8. Applied rewrites 88.4%

      \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(a \cdot c, \frac{-0.375}{b \cdot b}, -0.5\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{\mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (fma (* c a) (/ -0.375 (* b b)) -0.5) b)))

C

double code(double a, double b, double c) {
	return c * (fma((c * a), (-0.375 / (b * b)), -0.5) / b);
}

Julia

function code(a, b, c)
	return Float64(c * Float64(fma(Float64(c * a), Float64(-0.375 / Float64(b * b)), -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(c * a), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 78.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

   2. unpow3 N/A

      \[\leadsto c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{\color{blue}{\left(b \cdot b\right) \cdot b}} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   3. unpow2 N/A

      \[\leadsto c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}} \cdot b} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   4. associate-/r* N/A

      \[\leadsto c \cdot \left(\frac{-3}{8} \cdot \color{blue}{\frac{\frac{a \cdot c}{{b}^{2}}}{b}} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   5. associate-/l* N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}}{b}} - \frac{1}{2} \cdot \frac{1}{b}\right) \]

   6. associate-*r/ N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}}{b} - \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right) \]

   7. metadata-eval N/A

      \[\leadsto c \cdot \left(\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}}{b} - \frac{\color{blue}{\frac{1}{2}}}{b}\right) \]

   8. div-sub N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]

   9. lower-/.f64 N/A

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]

8. Applied rewrites 78.5%

   \[\leadsto \color{blue}{c \cdot \frac{\mathsf{fma}\left(a \cdot c, \frac{-0.375}{b \cdot b}, -0.5\right)}{b}} \]

9. Final simplification 78.5%

   \[\leadsto c \cdot \frac{\mathsf{fma}\left(c \cdot a, \frac{-0.375}{b \cdot b}, -0.5\right)}{b} \]
10. Add Preprocessing

Alternative 10: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return -0.5 * (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 = (-0.5d0) * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 63.5

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Applied rewrites 63.5%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))