Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.4%

Time: 14.6s

Alternatives: 5

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ t_2 := b \cdot t\_1\\ \mathsf{fma}\left(a, \mathsf{fma}\left(-c, \frac{c}{t\_1}, a \cdot \mathsf{fma}\left(-2, \frac{t\_0}{b \cdot t\_2}, \frac{-5 \cdot \left(a \cdot \left(c \cdot t\_0\right)\right)}{t\_1 \cdot t\_2}\right)\right), \frac{c}{-b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* b (* b b))) (t_2 (* b t_1)))
   (fma
    a
    (fma
     (- c)
     (/ c t_1)
     (*
      a
      (fma -2.0 (/ t_0 (* b t_2)) (/ (* -5.0 (* a (* c t_0))) (* t_1 t_2)))))
    (/ c (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * b);
	double t_2 = b * t_1;
	return fma(a, fma(-c, (c / t_1), (a * fma(-2.0, (t_0 / (b * t_2)), ((-5.0 * (a * (c * t_0))) / (t_1 * t_2))))), (c / -b));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(b * Float64(b * b))
	t_2 = Float64(b * t_1)
	return fma(a, fma(Float64(-c), Float64(c / t_1), Float64(a * fma(-2.0, Float64(t_0 / Float64(b * t_2)), Float64(Float64(-5.0 * Float64(a * Float64(c * t_0))) / Float64(t_1 * t_2))))), Float64(c / Float64(-b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * t$95$1), $MachinePrecision]}, N[(a * N[((-c) * N[(c / t$95$1), $MachinePrecision] + N[(a * N[(-2.0 * N[(t$95$0 / N[(b * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(a * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 95.6%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

      \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, -1 \cdot \frac{c}{b}\right)} \]

8. Applied rewrites 95.7%

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

10. Applied rewrites 95.7%

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

11. Final simplification 95.7%

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

Alternative 2: 93.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 95.6%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

      \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, -1 \cdot \frac{c}{b}\right)} \]

8. Applied rewrites 95.7%

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

9. Taylor expanded in b around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. lower-/.f64 N/A

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

11. Applied rewrites 94.3%

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

12. Taylor expanded in c around 0

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 94.3

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

14. Applied rewrites 94.3%

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

15. Final simplification 94.3%

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

Alternative 3: 90.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 95.6%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. lower-neg.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 91.8

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

8. Applied rewrites 91.8%

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

Alternative 4: 90.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.0%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.8

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

5. Applied rewrites 91.8%

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

6. Final simplification 91.8%

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

Alternative 5: 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 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 82.2

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

5. Applied rewrites 82.2%

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

Reproduce

herbie shell --seed 2024214 
(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)))