Quadratic roots, medium range

Percentage Accurate: 31.9% → 95.1%

Time: 11.8s

Alternatives: 6

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: 31.9% 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: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(a \cdot -0.25\right) \cdot \frac{{c}^{4} \cdot 20}{{b}^{7}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), -\frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (fma
   a
   (fma
    -2.0
    (/ (* c (* c c)) (pow b 5.0))
    (* (* a -0.25) (/ (* (pow c 4.0) 20.0) (pow b 7.0))))
   (/ (* c c) (* b (* b (- b)))))
  (- (/ c b))))

C

double code(double a, double b, double c) {
	return fma(a, fma(a, fma(-2.0, ((c * (c * c)) / pow(b, 5.0)), ((a * -0.25) * ((pow(c, 4.0) * 20.0) / pow(b, 7.0)))), ((c * c) / (b * (b * -b)))), -(c / b));
}

Julia

function code(a, b, c)
	return fma(a, fma(a, fma(-2.0, Float64(Float64(c * Float64(c * c)) / (b ^ 5.0)), Float64(Float64(a * -0.25) * Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 7.0)))), Float64(Float64(c * c) / Float64(b * Float64(b * Float64(-b))))), Float64(-Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(a * N[(a * N[(-2.0 * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.25), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(c / b), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in b around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 33.8

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

5. Simplified 33.8%

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

6. Taylor expanded in a around 0

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

8. Simplified 94.9%

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

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(\frac{-1}{4} \cdot a\right) \cdot \color{blue}{\left(20 \cdot \frac{{c}^{4}}{{b}^{7}}\right)}\right), \frac{c \cdot c}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot b\right) \cdot b}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(\frac{-1}{4} \cdot a\right) \cdot \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{7}}}\right), \frac{c \cdot c}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot b\right) \cdot b}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    2. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(\frac{-1}{4} \cdot a\right) \cdot \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{7}}}\right), \frac{c \cdot c}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot b\right) \cdot b}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(\frac{-1}{4} \cdot a\right) \cdot \frac{\color{blue}{{c}^{4} \cdot 20}}{{b}^{7}}\right), \frac{c \cdot c}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot b\right) \cdot b}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(\frac{-1}{4} \cdot a\right) \cdot \frac{\color{blue}{{c}^{4} \cdot 20}}{{b}^{7}}\right), \frac{c \cdot c}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot b\right) \cdot b}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(\frac{-1}{4} \cdot a\right) \cdot \frac{\color{blue}{{c}^{4}} \cdot 20}{{b}^{7}}\right), \frac{c \cdot c}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot b\right) \cdot b}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    6. lower-pow.f64 94.9

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(-0.25 \cdot a\right) \cdot \frac{{c}^{4} \cdot 20}{\color{blue}{{b}^{7}}}\right), \frac{c \cdot c}{\left(\left(-b\right) \cdot b\right) \cdot b}\right), \frac{c}{-b}\right) \]

11. Simplified 94.9%

    \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(-0.25 \cdot a\right) \cdot \color{blue}{\frac{{c}^{4} \cdot 20}{{b}^{7}}}\right), \frac{c \cdot c}{\left(\left(-b\right) \cdot b\right) \cdot b}\right), \frac{c}{-b}\right) \]

12. Final simplification 94.9%

    \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \left(a \cdot -0.25\right) \cdot \frac{{c}^{4} \cdot 20}{{b}^{7}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), -\frac{c}{b}\right) \]
13. Add Preprocessing

Alternative 2: 95.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.25 \cdot \frac{{c}^{4} \cdot \left(20 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{6}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (* (* a a) (/ (* c (* c c)) (* (* b b) (* b b))))
   (-
    (* -0.25 (/ (* (pow c 4.0) (* 20.0 (* a (* a a)))) (pow b 6.0)))
    (fma a (/ (* c c) (* b b)) c)))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, ((a * a) * ((c * (c * c)) / ((b * b) * (b * b)))), ((-0.25 * ((pow(c, 4.0) * (20.0 * (a * (a * a)))) / pow(b, 6.0))) - fma(a, ((c * c) / (b * b)), c))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64(Float64(a * a) * Float64(Float64(c * Float64(c * c)) / Float64(Float64(b * b) * Float64(b * b)))), Float64(Float64(-0.25 * Float64(Float64((c ^ 4.0) * Float64(20.0 * Float64(a * Float64(a * a)))) / (b ^ 6.0))) - fma(a, Float64(Float64(c * c) / Float64(b * b)), c))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(20.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in b around inf

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

5. Simplified 94.9%

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

6. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-1}{4} \cdot \color{blue}{\left(20 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-1}{4} \cdot \color{blue}{\frac{20 \cdot \left({a}^{3} \cdot {c}^{4}\right)}{{b}^{6}}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-1}{4} \cdot \color{blue}{\frac{20 \cdot \left({a}^{3} \cdot {c}^{4}\right)}{{b}^{6}}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-1}{4} \cdot \frac{\color{blue}{\left(20 \cdot {a}^{3}\right) \cdot {c}^{4}}}{{b}^{6}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-1}{4} \cdot \frac{\color{blue}{\left(20 \cdot {a}^{3}\right) \cdot {c}^{4}}}{{b}^{6}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-1}{4} \cdot \frac{\color{blue}{\left(20 \cdot {a}^{3}\right)} \cdot {c}^{4}}{{b}^{6}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-pow.f64 94.9

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

8. Simplified 94.9%

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

9. Final simplification 94.9%

   \[\leadsto \frac{\mathsf{fma}\left(-2, \left(a \cdot a\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.25 \cdot \frac{{c}^{4} \cdot \left(20 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{6}} - \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)\right)}{b} \]
10. Add Preprocessing

Alternative 3: 93.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 93.4%

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

6. Final simplification 93.4%

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

Alternative 4: 90.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 89.8

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

5. Simplified 89.8%

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

6. Final simplification 89.8%

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

Alternative 5: 89.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 89.4

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

5. Simplified 89.4%

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

6. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

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

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

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

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

   6. mul-1-neg N/A

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

   7. lower-*.f64 N/A

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

8. Simplified 89.5%

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

9. Taylor expanded in c around 0

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

11. Simplified 89.5%

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

Alternative 6: 80.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. lower-neg.f64 N/A

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

   3. lower-/.f64 79.9

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

5. Simplified 79.9%

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

Reproduce

herbie shell --seed 2024215 
(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)))