Quadratic roots, medium range

Percentage Accurate: 31.2% → 95.3%

Time: 12.0s

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: 31.2% 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: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({a}^{3} \cdot -5, \left({b}^{-6} \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right), \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \left({b}^{-4} \cdot c\right) \cdot \left(c \cdot c\right), \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* (pow a 3.0) -5.0)
   (* (* (pow b -6.0) (* c c)) (* c c))
   (fma
    (fma (* -2.0 a) (* (* (pow b -4.0) c) (* c c)) (/ (* (- c) c) (* b b)))
    a
    (- c)))
  b))

C

double code(double a, double b, double c) {
	return fma((pow(a, 3.0) * -5.0), ((pow(b, -6.0) * (c * c)) * (c * c)), fma(fma((-2.0 * a), ((pow(b, -4.0) * c) * (c * c)), ((-c * c) / (b * b))), a, -c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64((a ^ 3.0) * -5.0), Float64(Float64((b ^ -6.0) * Float64(c * c)) * Float64(c * c)), fma(fma(Float64(-2.0 * a), Float64(Float64((b ^ -4.0) * c) * Float64(c * c)), Float64(Float64(Float64(-c) * c) / Float64(b * b))), a, Float64(-c))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * -5.0), $MachinePrecision] * N[(N[(N[Power[b, -6.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * a), $MachinePrecision] * N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[((-c) * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + (-c)), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 94.5%

   \[\leadsto \frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, \frac{{c}^{4}}{{b}^{6}}, \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{{c}^{3}}{{b}^{4}}, \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)\right)}{\color{blue}{b}} \]
9. Step-by-step derivation

10. Applied rewrites 94.5%

    \[\leadsto \frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, \frac{{c}^{4}}{{b}^{6}}, \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \left(c \cdot c\right) \cdot \left(c \cdot {b}^{-4}\right), \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)\right)}{b} \]
11. Step-by-step derivation

12. Applied rewrites 94.5%

    \[\leadsto \frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, \left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right), \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \left(c \cdot c\right) \cdot \left(c \cdot {b}^{-4}\right), \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)\right)}{b} \]

13. Final simplification 94.5%

    \[\leadsto \frac{\mathsf{fma}\left({a}^{3} \cdot -5, \left({b}^{-6} \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right), \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \left({b}^{-4} \cdot c\right) \cdot \left(c \cdot c\right), \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)\right)}{b} \]
14. Add Preprocessing

Alternative 2: 95.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-1, \frac{a}{b \cdot b}, \mathsf{fma}\left(-5, \frac{{a}^{3} \cdot c}{{b}^{6}}, \frac{a \cdot a}{{b}^{4}} \cdot -2\right) \cdot c\right), -1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   (fma
    c
    (fma
     -1.0
     (/ a (* b b))
     (*
      (fma
       -5.0
       (/ (* (pow a 3.0) c) (pow b 6.0))
       (* (/ (* a a) (pow b 4.0)) -2.0))
      c))
    -1.0)
   c)
  b))

C

double code(double a, double b, double c) {
	return (fma(c, fma(-1.0, (a / (b * b)), (fma(-5.0, ((pow(a, 3.0) * c) / pow(b, 6.0)), (((a * a) / pow(b, 4.0)) * -2.0)) * c)), -1.0) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(c, fma(-1.0, Float64(a / Float64(b * b)), Float64(fma(-5.0, Float64(Float64((a ^ 3.0) * c) / (b ^ 6.0)), Float64(Float64(Float64(a * a) / (b ^ 4.0)) * -2.0)) * c)), -1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(-1.0 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 94.5%

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

9. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 1\right)}{b} \]
10. Step-by-step derivation

11. Applied rewrites 94.4%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-1, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-5, \frac{{a}^{3} \cdot c}{{b}^{6}}, -2 \cdot \frac{a \cdot a}{{b}^{4}}\right)\right), -1\right)}{b} \]

12. Final simplification 94.4%

    \[\leadsto \frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-1, \frac{a}{b \cdot b}, \mathsf{fma}\left(-5, \frac{{a}^{3} \cdot c}{{b}^{6}}, \frac{a \cdot a}{{b}^{4}} \cdot -2\right) \cdot c\right), -1\right) \cdot c}{b} \]
13. Add Preprocessing

Alternative 3: 93.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{{c}^{3}}{{b}^{4}}, \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (fma (* -2.0 a) (/ (pow c 3.0) (pow b 4.0)) (/ (* (- c) c) (* b b)))
   a
   (- c))
  b))

C

double code(double a, double b, double c) {
	return fma(fma((-2.0 * a), (pow(c, 3.0) / pow(b, 4.0)), ((-c * c) / (b * b))), a, -c) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(fma(Float64(-2.0 * a), Float64((c ^ 3.0) / (b ^ 4.0)), Float64(Float64(Float64(-c) * c) / Float64(b * b))), a, Float64(-c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[((-c) * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + (-c)), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 92.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{{c}^{3}}{{b}^{4}}, \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)}{b}} \]
9. Add Preprocessing

Alternative 4: 93.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Applied rewrites 92.9%

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

7. Applied rewrites 92.9%

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

8. Final simplification 92.9%

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

Alternative 5: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot \left(\left(a \cdot a\right) \cdot -2\right) - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (fma (- (* (/ c (pow b 4.0)) (* (* a a) -2.0)) (/ a (* b b))) c -1.0) c)
  b))

C

double code(double a, double b, double c) {
	return (fma((((c / pow(b, 4.0)) * ((a * a) * -2.0)) - (a / (b * b))), c, -1.0) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(Float64(Float64(c / (b ^ 4.0)) * Float64(Float64(a * a) * -2.0)) - Float64(a / Float64(b * b))), c, -1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 32.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Applied rewrites 92.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 92.8%

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

9. Final simplification 92.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot \left(\left(a \cdot a\right) \cdot -2\right) - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \]
10. Add Preprocessing

Alternative 6: 90.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-c - (((c * c) * a) / (b * b))) / 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 - (((c * c) * a) / (b * b))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 94.5%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 89.9%

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

12. Final simplification 89.9%

    \[\leadsto \frac{\left(-c\right) - \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}}{b} \]
13. Add Preprocessing

Alternative 7: 90.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Applied rewrites 92.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 89.8%

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

Alternative 8: 81.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 80.3

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

5. Applied rewrites 80.3%

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

Reproduce

herbie shell --seed 2024331 
(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)))