Quadratic roots, wide range

Percentage Accurate: 18.0% → 99.7%

Time: 13.4s

Alternatives: 7

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 18.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return Float64(Float64(fma(c, Float64(a * -4.0), 0.0) / 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[(c * N[(a * -4.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.2%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sub-neg N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval 19.2

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

4. Applied egg-rr 19.2%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. --lowering--.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. +-commutative N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 19.4

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

6. Applied egg-rr 19.4%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. associate--l+ N/A

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

   6. +-inverses N/A

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot -4\right) + \color{blue}{0}}{2 \cdot a}}{b + \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}} \]

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 99.8

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

8. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sub-neg N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval 19.2

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

4. Applied egg-rr 19.2%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. --lowering--.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. +-commutative N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 19.4

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

6. Applied egg-rr 19.4%

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

7. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-lowering-*.f64 99.5

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * -4.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * N[(2.0 * 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 19.2%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sub-neg N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval 19.2

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

4. Applied egg-rr 19.2%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. --lowering--.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. +-commutative N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 19.4

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

6. Applied egg-rr 19.4%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{c \cdot \left(a \cdot -4\right) + \color{blue}{0}}{\left(2 \cdot a\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}\right)} \]

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 99.4

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

8. Applied egg-rr 99.4%

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

Alternative 4: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return Float64(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 19.2%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
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. neg-lowering-neg.f64 N/A

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

   5. /-lowering-/.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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.f64 95.0

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

5. Simplified 95.0%

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

6. Final simplification 95.0%

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

Alternative 5: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Simplified 98.2%

   \[\leadsto \color{blue}{\left(-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\right) + \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{{c}^{4} \cdot 20}{{b}^{6}} \cdot \frac{a}{b}, -0.25, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -2}{{b}^{5}}\right)} \]

6. Taylor expanded in c around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. associate-/l* N/A

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

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

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

   6. neg-mul-1 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. neg-lowering-neg.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. distribute-neg-frac N/A

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

   17. metadata-eval N/A

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

   18. /-lowering-/.f64 94.7

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

8. Simplified 94.7%

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

9. Taylor expanded in b around 0

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

    1. /-lowering-/.f64 N/A

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

    2. distribute-lft-out N/A

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

    3. mul-1-neg N/A

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

    4. neg-lowering-neg.f64 N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. cube-mult N/A

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 N/A

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

    11. unpow2 N/A

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

    12. *-lowering-*.f64 94.6

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

11. Simplified 94.6%

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

12. Final simplification 94.6%

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

Alternative 6: 90.3% 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 19.2%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 89.3

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

5. Simplified 89.3%

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

6. Final simplification 89.3%

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

Alternative 7: 3.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 19.2%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sub-neg N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 N/A

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

   13. metadata-eval 19.2

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

4. Applied egg-rr 19.2%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a} \]

   4. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a} \]

   6. neg-sub0 N/A

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(b \cdot b\right)}}{0 + b} + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a} \]

   7. pow1/2 N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + \color{blue}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{1}{2}}}}{2 \cdot a} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + {\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{\left(\frac{-1}{2} + 1\right)}}}{2 \cdot a} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + {\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + 1\right)}}{2 \cdot a} \]

   10. pow-plus N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + \color{blue}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}{2 \cdot a} \]

   11. pow-flip N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + \color{blue}{\frac{1}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{1}{2}}}} \cdot \left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}{2 \cdot a} \]

   12. pow1/2 N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + \frac{1}{\color{blue}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}} \cdot \left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}{2 \cdot a} \]

   13. associate-/r/ N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b} + \color{blue}{\frac{1}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{2 \cdot a} \]

   14. frac-add N/A

       \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b \cdot b\right)\right) \cdot \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{b \cdot b + c \cdot \left(a \cdot -4\right)} + \left(0 + b\right) \cdot 1}{\left(0 + b\right) \cdot \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{2 \cdot a} \]

6. Applied egg-rr 20.3%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b \cdot \left(-b\right), \frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}, \left(0 + b\right) \cdot 1\right)}{\left(0 + b\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}}{2 \cdot a} \]

7. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. /-lowering-/.f64 3.3

      \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))