Quadratic roots, wide range

Percentage Accurate: 17.9% → 97.6%

Time: 11.5s

Alternatives: 7

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.9% 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: 97.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 97.6%

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

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 97.6%

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

13. Applied rewrites 97.6%

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

14. Final simplification 97.6%

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

Alternative 2: 96.8% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (/ (* (* (* (pow c 3.0) a) 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 (((((pow(c, 3.0) * a) * 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((c ^ 3.0) * a) * 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[Power[c, 3.0], $MachinePrecision] * a), $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 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 96.4%

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

Alternative 3: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 18.1%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 96.2%

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

8. Taylor expanded in b around inf

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

10. Applied rewrites 96.2%

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

11. Final simplification 96.2%

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

Alternative 4: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-/l* N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 94.6%

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

6. Final simplification 94.6%

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

Alternative 5: 95.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

4. Applied rewrites 18.1%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 94.4

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

7. Applied rewrites 94.4%

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

8. Final simplification 94.4%

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

Alternative 6: 90.3% 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 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 89.8

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

5. Applied rewrites 89.8%

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

Alternative 7: 3.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 18.1%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 18.2

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

4. Applied rewrites 18.2%

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

6. Applied rewrites 19.5%

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

7. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. lower-/.f64 3.3

      \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Taylor expanded in a around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 3.3%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))