Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.9%

Time: 9.5s

Alternatives: 4

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: 17.6% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 18.3

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

4. Applied rewrites 18.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 18.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 18.8

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

   7. lift--.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. associate--l+ N/A

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

   10. +-inverses N/A

       \[\leadsto \frac{\frac{\left(-4 \cdot c\right) \cdot a + \color{blue}{0}}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - \left(-b\right)} \]

   11. lower-fma.f64 99.8

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - \left(-b\right)} \]

   12. lift--.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{2 \cdot a}}{\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - \left(-b\right)}} \]

   13. sub-neg N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{2 \cdot a}}{\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(\mathsf{neg}\left(\left(-b\right)\right)\right)}} \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

9. Taylor expanded in a around 0

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

    1. lower-*.f64 99.9

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

11. Applied rewrites 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 95.2% 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 18.2%

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

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 18.2

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \]

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 18.2

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

4. Applied rewrites 18.2%

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

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 93.6

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

7. Applied rewrites 93.6%

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

8. Final simplification 93.6%

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

Alternative 3: 95.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-/.f64 93.5

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

5. Applied rewrites 93.5%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 93.5%

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

9. Final simplification 93.5%

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

Alternative 4: 90.6% 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 18.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 89.7

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

5. Applied rewrites 89.7%

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

Reproduce

herbie shell --seed 2024285 
(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)))