Quadratic roots, wide range

Percentage Accurate: 18.0% → 99.9%

Time: 8.4s

Alternatives: 5

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: 18.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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 18.2%

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

6. Applied rewrites 18.6%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(c * -2.0), $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * -2.0 + N[(2.0 * b), $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. 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 18.2%

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

6. Applied rewrites 18.6%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 95.3

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

12. Applied rewrites 95.3%

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

13. Final simplification 95.3%

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

Alternative 3: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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 18.2%

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

6. Applied rewrites 18.6%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 95.2

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

12. Applied rewrites 95.2%

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

13. Final simplification 95.2%

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

Alternative 4: 95.1% 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 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 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 96.7%

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

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 95.1%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 95.1%

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

Alternative 5: 90.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 90.2

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

5. Applied rewrites 90.2%

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

Reproduce

herbie shell --seed 2024278 
(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)))