Quadratic roots, narrow range

Percentage Accurate: 55.4% → 99.5%

Time: 8.2s

Alternatives: 8

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{\mathsf{fma}\left(a \cdot -4, c, 0\right)}, a, \frac{b}{c} \cdot -0.25\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (fma
   (/ (sqrt (fma (* a -4.0) c (* b b))) (fma (* a -4.0) c 0.0))
   a
   (* (/ b c) -0.25))))

C

double code(double a, double b, double c) {
	return 0.5 / fma((sqrt(fma((a * -4.0), c, (b * b))) / fma((a * -4.0), c, 0.0)), a, ((b / c) * -0.25));
}

Julia

function code(a, b, c)
	return Float64(0.5 / fma(Float64(sqrt(fma(Float64(a * -4.0), c, Float64(b * b))) / fma(Float64(a * -4.0), c, 0.0)), a, Float64(Float64(b / c) * -0.25)))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(a * -4.0), $MachinePrecision] * c + 0.0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 52.6%

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

6. Applied rewrites 53.9%

   \[\leadsto \frac{0.5}{\color{blue}{\left({\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right)}^{-1} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)\right)} \cdot a} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

   \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a\right)}} \]

9. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{1}{2}}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, \color{blue}{\frac{-1}{4} \cdot \frac{b}{c}}\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, \color{blue}{\frac{-1}{4} \cdot \frac{b}{c}}\right)} \]

    2. lower-/.f64 99.4

       \[\leadsto \frac{0.5}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, -0.25 \cdot \color{blue}{\frac{b}{c}}\right)} \]

11. Applied rewrites 99.4%

    \[\leadsto \frac{0.5}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, \color{blue}{-0.25 \cdot \frac{b}{c}}\right)} \]

12. Final simplification 99.4%

    \[\leadsto \frac{0.5}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{\mathsf{fma}\left(a \cdot -4, c, 0\right)}, a, \frac{b}{c} \cdot -0.25\right)} \]
13. Add Preprocessing

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{a}}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + b\right) \cdot \frac{-0.25}{c \cdot a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ 0.5 a) (* (+ (sqrt (fma c (* a -4.0) (* b b))) b) (/ -0.25 (* c a)))))

C

double code(double a, double b, double c) {
	return (0.5 / a) / ((sqrt(fma(c, (a * -4.0), (b * b))) + b) * (-0.25 / (c * a)));
}

Julia

function code(a, b, c)
	return Float64(Float64(0.5 / a) / Float64(Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) + b) * Float64(-0.25 / Float64(c * a))))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.5 / a), $MachinePrecision] / N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(-0.25 / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 52.6%

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

6. Applied rewrites 53.9%

   \[\leadsto \frac{0.5}{\color{blue}{\left({\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right)}^{-1} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)\right)} \cdot a} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

   \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a\right)}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}, a, \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a\right)}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a + \color{blue}{\frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} \cdot a}} \]

   4. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{a \cdot \left(\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{a}}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{a}}}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{a}}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}}} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{2}}{a}}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} \]

   9. div-inv N/A

      \[\leadsto \frac{\frac{\frac{1}{2}}{a}}{\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \frac{1}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} + \frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \frac{1}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} + \color{blue}{\frac{b}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}}} \]

   11. div-inv N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \frac{1}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)} + \color{blue}{b \cdot \frac{1}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}}} \]

10. Applied rewrites 99.2%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{-0.25}{c \cdot a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + b\right)}} \]

11. Final simplification 99.2%

    \[\leadsto \frac{\frac{0.5}{a}}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + b\right) \cdot \frac{-0.25}{c \cdot a}} \]
12. Add Preprocessing

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} + b\right) \cdot a}{\mathsf{fma}\left(a \cdot -4, c, 0\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (/ (* (+ (sqrt (fma (* a -4.0) c (* b b))) b) a) (fma (* a -4.0) c 0.0))))

C

double code(double a, double b, double c) {
	return 0.5 / (((sqrt(fma((a * -4.0), c, (b * b))) + b) * a) / fma((a * -4.0), c, 0.0));
}

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(Float64(Float64(sqrt(fma(Float64(a * -4.0), c, Float64(b * b))) + b) * a) / fma(Float64(a * -4.0), c, 0.0)))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision] / N[(N[(a * -4.0), $MachinePrecision] * c + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 52.6%

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

6. Applied rewrites 53.9%

   \[\leadsto \frac{0.5}{\color{blue}{\left({\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right)}^{-1} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)\right)} \cdot a} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow-1 N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 53.9

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

   10. lift--.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. associate--l+ N/A

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

8. Applied rewrites 99.2%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right) \cdot a\right)}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}}} \]

9. Final simplification 99.2%

   \[\leadsto \frac{0.5}{\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} + b\right) \cdot a}{\mathsf{fma}\left(a \cdot -4, c, 0\right)}} \]
10. Add Preprocessing

Alternative 4: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\left(\frac{-0.25}{c \cdot a} \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} + b\right)\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.5 (* (* (/ -0.25 (* c a)) (+ (sqrt (fma (* a -4.0) c (* b b))) b)) a)))

C

double code(double a, double b, double c) {
	return 0.5 / (((-0.25 / (c * a)) * (sqrt(fma((a * -4.0), c, (b * b))) + b)) * a);
}

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(Float64(Float64(-0.25 / Float64(c * a)) * Float64(sqrt(fma(Float64(a * -4.0), c, Float64(b * b))) + b)) * a))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(N[(-0.25 / N[(c * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 52.6%

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

6. Applied rewrites 53.9%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.2

      \[\leadsto \frac{0.5}{\left(\frac{-0.25}{\color{blue}{a \cdot c}} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)\right) \cdot a} \]

9. Applied rewrites 99.2%

   \[\leadsto \frac{0.5}{\left(\color{blue}{\frac{-0.25}{a \cdot c}} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)\right) \cdot a} \]

10. Final simplification 99.2%

    \[\leadsto \frac{0.5}{\left(\frac{-0.25}{c \cdot a} \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} + b\right)\right) \cdot a} \]
11. Add Preprocessing

Alternative 5: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.5 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 18.5)
   (* (- (sqrt (fma b b (* (* a -4.0) c))) b) (/ 0.5 a))
   (/ 0.5 (/ (fma 0.5 (* (/ c b) a) (* -0.5 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 18.5) {
		tmp = (sqrt(fma(b, b, ((a * -4.0) * c))) - b) * (0.5 / a);
	} else {
		tmp = 0.5 / (fma(0.5, ((c / b) * a), (-0.5 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 18.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -4.0) * c))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(0.5 / Float64(fma(0.5, Float64(Float64(c / b) * a), Float64(-0.5 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 18.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 79.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.3

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

   4. Applied rewrites 79.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. +-rgt-identity N/A

         \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\sqrt{b \cdot b + \color{blue}{\left(\left(-4 \cdot a\right) \cdot c + 0\right)}} - b\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\sqrt{b \cdot b + \color{blue}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} - b\right) \]

      11. lower-fma.f64 79.4

          \[\leadsto \frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, 0\right)\right)}} - b\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c + 0}\right)} - b\right) \]

      13. +-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.4

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

   4. Applied rewrites 43.5%

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

   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. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(0.5, \frac{c}{b} \cdot a, -0.5 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, -0.5 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 18.5)
   (* (- (sqrt (fma b b (* (* a -4.0) c))) b) (/ 0.5 a))
   (/ 0.5 (fma (/ a b) 0.5 (* -0.5 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 18.5) {
		tmp = (sqrt(fma(b, b, ((a * -4.0) * c))) - b) * (0.5 / a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, (-0.5 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 18.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -4.0) * c))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(-0.5 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 18.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 79.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.3

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

   4. Applied rewrites 79.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. +-rgt-identity N/A

         \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\sqrt{b \cdot b + \color{blue}{\left(\left(-4 \cdot a\right) \cdot c + 0\right)}} - b\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\sqrt{b \cdot b + \color{blue}{\mathsf{fma}\left(-4 \cdot a, c, 0\right)}} - b\right) \]

      11. lower-fma.f64 79.4

          \[\leadsto \frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, 0\right)\right)}} - b\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c + 0}\right)} - b\right) \]

      13. +-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.4

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.4

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

   6. Applied rewrites 79.4%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

   4. Applied rewrites 43.5%

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

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.5

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

   7. Applied rewrites 90.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, -0.5 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 52.6%

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

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 83.4

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

7. Applied rewrites 83.4%

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

8. Final simplification 83.4%

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

Alternative 8: 64.4% 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 52.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 66.8

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

5. Applied rewrites 66.8%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))