Quadratic roots, medium range

Percentage Accurate: 31.9% → 99.7%

Time: 12.1s

Alternatives: 6

Speedup: 3.6×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.9% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. lift-*.f64 N/A

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

   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 33.9

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

4. Applied egg-rr 33.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. flip-+ N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqr-neg N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

6. Applied egg-rr 34.3%

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

8. Applied egg-rr 99.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift--.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. lift-*.f64 N/A

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

   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 33.9

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

4. Applied egg-rr 33.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. flip-+ N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqr-neg N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

6. Applied egg-rr 34.3%

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

8. Applied egg-rr 99.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.4

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. lift-*.f64 N/A

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

   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 33.9

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

4. Applied egg-rr 33.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. flip-+ N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqr-neg N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. rem-square-sqrt N/A

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

6. Applied egg-rr 34.3%

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

8. Applied egg-rr 34.7%

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

9. Taylor expanded in b around 0

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

    1. lower-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 99.3

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 4: 90.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in c around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. cube-mult N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-/.f64 89.5

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

5. Simplified 89.5%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 89.8

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

8. Simplified 89.8%

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

Alternative 5: 90.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 89.8

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

5. Simplified 89.8%

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

6. Final simplification 89.8%

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

Alternative 6: 80.9% 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(c / Float64(-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 33.8%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 79.9

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

5. Simplified 79.9%

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

6. Final simplification 79.9%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))