Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.7%

Time: 13.9s

Alternatives: 6

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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 15.1

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

4. Applied egg-rr 15.1%

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

6. Applied egg-rr 99.5%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 15.1

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

4. Applied egg-rr 15.1%

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

6. Applied egg-rr 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 95.3% 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(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-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 15.0%

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

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.4

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

5. Simplified 95.4%

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

6. Final simplification 95.4%

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

Alternative 4: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in a around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 95.3

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

5. Simplified 95.3%

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

6. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 95.0

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

8. Simplified 95.0%

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

9. Taylor expanded in c around 0

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

    1. lower-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 95.0

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

11. Simplified 95.0%

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

12. Final simplification 95.0%

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

Alternative 5: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in a around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 95.3

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

5. Simplified 95.3%

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

6. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 95.0

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

8. Simplified 95.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 95.0

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

10. Applied egg-rr 95.0%

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

11. Final simplification 95.0%

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

Alternative 6: 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 15.0%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 92.1

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

5. Simplified 92.1%

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

6. Final simplification 92.1%

   \[\leadsto -\frac{c}{b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))