Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.9%

Time: 10.4s

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: 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 15.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 15.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 15.8

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

4. Applied rewrites 15.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 16.2%

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

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 99.8

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Final simplification 97.2%

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

Alternative 3: 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 15.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 97.1%

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

9. Final simplification 97.1%

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

Alternative 4: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 96.7%

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

10. Applied rewrites 96.7%

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

11. Final simplification 96.7%

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

Alternative 5: 90.5% 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 15.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 92.1

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

5. Applied rewrites 92.1%

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

Reproduce

herbie shell --seed 2024235 
(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)))