Quadratic roots, medium range

Percentage Accurate: 31.7% → 95.5%

Time: 13.4s

Alternatives: 8

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \left(-2 \cdot {b}^{-5}\right), \frac{-0.25 \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{t\_0 \cdot \left(b \cdot t\_0\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (fma
     c
     (* (* c c) (* -2.0 (pow b -5.0)))
     (/ (* -0.25 (* a (* (* c c) (* (* c c) 20.0)))) (* t_0 (* b t_0))))
    (* a a)
    (/ (fma (* c c) (/ a (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(fma(c, ((c * c) * (-2.0 * pow(b, -5.0))), ((-0.25 * (a * ((c * c) * ((c * c) * 20.0)))) / (t_0 * (b * t_0)))), (a * a), (fma((c * c), (a / (b * b)), c) / -b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(fma(c, Float64(Float64(c * c) * Float64(-2.0 * (b ^ -5.0))), Float64(Float64(-0.25 * Float64(a * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0)))) / Float64(t_0 * Float64(b * t_0)))), Float64(a * a), Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \left(-2 \cdot {b}^{-5}\right), \frac{-0.25 \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right) \]
9. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \mathsf{fma}\left(a, -\frac{c \cdot c}{{b}^{5}}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{1}{b}\right), \frac{-b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   a
   (fma
    a
    (* -2.0 (fma a (- (/ (* c c) (pow b 5.0))) (* (/ c (* b (* b b))) -0.5)))
    (/ 1.0 b))
   (/ (- b) c))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma(a, (-2.0 * fma(a, -((c * c) / pow(b, 5.0)), ((c / (b * (b * b))) * -0.5))), (1.0 / b)), (-b / c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(a, Float64(-2.0 * fma(a, Float64(-Float64(Float64(c * c) / (b ^ 5.0))), Float64(Float64(c / Float64(b * Float64(b * b))) * -0.5))), Float64(1.0 / b)), Float64(Float64(-b) / c)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(a * N[(-2.0 * N[(a * (-N[(N[(c * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]) + N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[((-b) / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

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

7. Applied rewrites 31.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 96.4%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.125, \frac{b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)}{c \cdot c}, \frac{c \cdot c}{{b}^{5}}\right) - \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{1}{b}\right), \frac{b}{-c}\right)}} \]

11. Taylor expanded in b around 0

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

    1. distribute-lft1-in N/A

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

    2. metadata-eval N/A

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

    3. distribute-rgt-out-- N/A

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

    4. metadata-eval N/A

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

    5. *-commutative N/A

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

    6. mul-1-neg N/A

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

    7. distribute-frac-neg N/A

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

    8. distribute-neg-frac2 N/A

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

    9. mul-1-neg N/A

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

    10. lower-/.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. mul-1-neg N/A

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

    14. lower-neg.f64 N/A

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

    15. lower-pow.f64 96.4

        \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \mathsf{fma}\left(a, \frac{c \cdot c}{-\color{blue}{{b}^{5}}}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{1}{b}\right), \frac{b}{-c}\right)} \]

13. Applied rewrites 96.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \mathsf{fma}\left(a, \color{blue}{\frac{c \cdot c}{-{b}^{5}}}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{1}{b}\right), \frac{b}{-c}\right)} \]

14. Final simplification 96.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -2 \cdot \mathsf{fma}\left(a, -\frac{c \cdot c}{{b}^{5}}, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{1}{b}\right), \frac{-b}{c}\right)} \]
15. Add Preprocessing

Alternative 3: 94.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

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

7. Applied rewrites 31.7%

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

8. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 94.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-2 \cdot c, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{a}{b}\right), -b\right)}{c}}} \]
11. Step-by-step derivation

12. Applied rewrites 95.2%

    \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot \frac{a}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \left(c \cdot -2\right), \frac{a}{b}\right), -b\right)}} \]
13. Add Preprocessing

Alternative 4: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1}{b}\right), \frac{-b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (fma a (fma a (/ c (* b (* b b))) (/ 1.0 b)) (/ (- b) c))))

C

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

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(a, Float64(c / Float64(b * Float64(b * b))), Float64(1.0 / b)), Float64(Float64(-b) / c)))
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

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

7. Applied rewrites 31.7%

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

8. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 94.9%

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

11. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. associate-/l* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. cube-mult N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. mul-1-neg N/A

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

    14. distribute-neg-frac2 N/A

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

    15. mul-1-neg N/A

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

    16. lower-/.f64 N/A

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

    17. mul-1-neg N/A

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

    18. lower-neg.f64 94.9

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

13. Applied rewrites 94.9%

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

14. Final simplification 94.9%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1}{b}\right), \frac{-b}{c}\right)} \]
15. Add Preprocessing

Alternative 5: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(c, \frac{\mathsf{fma}\left(a, a \cdot \frac{c}{b \cdot b}, a\right)}{b}, -b\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (fma c (/ (fma a (* a (/ c (* b b))) a) b) (- b)) c)))

C

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

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(c, Float64(fma(a, Float64(a * Float64(c / Float64(b * b))), a) / b), Float64(-b)) / c))
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

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

7. Applied rewrites 31.7%

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

8. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 94.9%

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

11. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. associate-/l* N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. associate-/l* N/A

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

    7. lower-fma.f64 N/A

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

    8. associate-/l* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 94.9

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

13. Applied rewrites 94.9%

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

Alternative 6: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

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

7. Applied rewrites 31.7%

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

8. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 91.1

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

10. Applied rewrites 91.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{c \cdot a}{b} - b}{c}}} \]
11. Add Preprocessing

Alternative 7: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Fortran

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

Java

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

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 31.8

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

5. Applied rewrites 31.8%

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

7. Applied rewrites 31.7%

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

8. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 91.1

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

10. Applied rewrites 91.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
11. Add Preprocessing

Alternative 8: 81.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 81.2

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

5. Applied rewrites 81.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))