Quadratic roots, wide range

Percentage Accurate: 18.3% → 99.4%

Time: 9.9s

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: 18.3% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 17.5

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

5. Applied rewrites 17.5%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f64 N/A

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

7. Applied rewrites 17.9%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/l/ N/A

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

    4. lower-/.f64 N/A

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

12. Applied rewrites 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a \cdot c, 4, 0\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma (* a c) 4.0 0.0)
  (* (- (- b) (sqrt (fma (* a c) -4.0 (* b b)))) (* 2.0 a))))

C

double code(double a, double b, double c) {
	return fma((a * c), 4.0, 0.0) / ((-b - sqrt(fma((a * c), -4.0, (b * b)))) * (2.0 * a));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(a * c), 4.0, 0.0) / Float64(Float64(Float64(-b) - sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))) * Float64(2.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * c), $MachinePrecision] * 4.0 + 0.0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.4%

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

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 17.4

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

4. Applied rewrites 17.4%

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

6. Applied rewrites 99.5%

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

Alternative 3: 95.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. div-add N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 95.1%

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

7. Applied rewrites 95.1%

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

8. Final simplification 95.1%

   \[\leadsto \frac{\frac{\frac{c \cdot c}{b} \cdot a}{b}}{-b} - \frac{c}{b} \]
9. Add Preprocessing

Alternative 4: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. div-add N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 95.1%

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

7. Applied rewrites 95.1%

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

Alternative 5: 95.2% 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(a, Float64(c / Float64(b * b)), 1.0) * c) / Float64(-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 17.4%

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

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. div-add N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 95.1%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 95.0%

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

9. Final simplification 95.0%

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

Alternative 6: 94.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 94.7%

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

Alternative 7: 90.1% 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 17.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 90.5

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

5. Applied rewrites 90.5%

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

Reproduce

herbie shell --seed 2024340 
(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)))