Quadratic roots, medium range

Percentage Accurate: 31.6% → 95.5%

Time: 12.1s

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: 31.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot c\\ t_1 := \frac{c \cdot c}{b \cdot b}\\ \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{b \cdot b} \cdot t\_1, \mathsf{fma}\left(a, \frac{t\_0 \cdot t\_0}{{b}^{6}} \cdot -5 - t\_1, -c\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) c)) (t_1 (/ (* c c) (* b b))))
   (/
    (fma
     (* (* -2.0 a) a)
     (* (/ c (* b b)) t_1)
     (fma a (- (* (/ (* t_0 t_0) (pow b 6.0)) -5.0) t_1) (- c)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * c;
	double t_1 = (c * c) / (b * b);
	return fma(((-2.0 * a) * a), ((c / (b * b)) * t_1), fma(a, ((((t_0 * t_0) / pow(b, 6.0)) * -5.0) - t_1), -c)) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * c)
	t_1 = Float64(Float64(c * c) / Float64(b * b))
	return Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(Float64(c / Float64(b * b)) * t_1), fma(a, Float64(Float64(Float64(Float64(t_0 * t_0) / (b ^ 6.0)) * -5.0) - t_1), Float64(-c))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(a * N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision] - t$95$1), $MachinePrecision] + (-c)), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 32.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.25}{a}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{{b}^{6}}, -\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, a \cdot \left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{6}} - \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 95.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(a, -5 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{4}}{{b}^{6}} - \frac{c \cdot c}{b \cdot b}, -c\right)\right)}{b} \]
11. Step-by-step derivation

12. Applied rewrites 95.9%

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

13. Final simplification 95.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{b \cdot b} \cdot \frac{c \cdot c}{b \cdot b}, \mathsf{fma}\left(a, \frac{\left(\left(c \cdot a\right) \cdot c\right) \cdot \left(\left(c \cdot a\right) \cdot c\right)}{{b}^{6}} \cdot -5 - \frac{c \cdot c}{b \cdot b}, -c\right)\right)}{b} \]
14. Add Preprocessing

Alternative 2: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, \mathsf{fma}\left(-2, \left(-0.5 \cdot \frac{c}{{b}^{3}}\right) \cdot a, \frac{1}{b}\right) \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   -1.0
   (/ b c)
   (* (fma -2.0 (* (* -0.5 (/ c (pow b 3.0))) a) (/ 1.0 b)) a))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-1.0, (b / c), (fma(-2.0, ((-0.5 * (c / pow(b, 3.0))) * a), (1.0 / b)) * a));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-1.0, Float64(b / c), Float64(fma(-2.0, Float64(Float64(-0.5 * Float64(c / (b ^ 3.0))) * a), Float64(1.0 / b)) * a)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision] + N[(N[(-2.0 * N[(N[(-0.5 * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.8

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

5. Applied rewrites 32.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 32.8

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 32.8

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

7. Applied rewrites 32.8%

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

8. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-/.f64 94.2

       \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, a \cdot \mathsf{fma}\left(-2, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), \color{blue}{\frac{1}{b}}\right)\right)} \]

10. Applied rewrites 94.2%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{b}{c}, a \cdot \mathsf{fma}\left(-2, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), \frac{1}{b}\right)\right)}} \]

11. Final simplification 94.2%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, \mathsf{fma}\left(-2, \left(-0.5 \cdot \frac{c}{{b}^{3}}\right) \cdot a, \frac{1}{b}\right) \cdot a\right)} \]
12. Add Preprocessing

Alternative 3: 93.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.25}{a}, \left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{{b}^{6}}, -\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)\right)\right)}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 94.2%

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

11. Final simplification 94.2%

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

Alternative 4: 90.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 32.8

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

5. Applied rewrites 32.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 32.8

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 32.8

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

7. Applied rewrites 32.8%

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

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. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 90.9

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

10. Applied rewrites 90.9%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)}} \]
11. 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(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 32.9%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 80.4

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

5. Applied rewrites 80.4%

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

Reproduce

herbie shell --seed 2024283 
(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)))