Quadratic roots, medium range

Percentage Accurate: 31.2% → 99.6%

Time: 14.8s

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: 31.2% 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: 99.6% 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 30.9%

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

4. Applied egg-rr 31.3%

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

   1. sub-div N/A

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

   2. associate-/l/ N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 31.6%

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

   1. associate-/r* N/A

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

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

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

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

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

   4. associate--l+ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. +-inverses N/A

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

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

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

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

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

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

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

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

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

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

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

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

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

8. Applied egg-rr 99.6%

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

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(c \cdot a, 4, 0\right) \cdot \frac{1}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot -2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma (* c a) 4.0 0.0)
  (/ 1.0 (* (+ b (sqrt (fma c (* a -4.0) (* b b)))) (* a -2.0)))))

C

double code(double a, double b, double c) {
	return fma((c * a), 4.0, 0.0) * (1.0 / ((b + sqrt(fma(c, (a * -4.0), (b * b)))) * (a * -2.0)));
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c * a), 4.0, 0.0) * Float64(1.0 / Float64(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[(c * a), $MachinePrecision] * 4.0 + 0.0), $MachinePrecision] * N[(1.0 / N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

4. Applied egg-rr 31.3%

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

   1. sub-div N/A

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

   2. associate-/l/ N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 31.6%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

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

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

   4. associate--l+ N/A

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

   5. +-inverses N/A

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

   6. distribute-neg-in N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f64 N/A

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

8. Applied egg-rr 99.4%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

4. Applied egg-rr 31.3%

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

   1. sub-div N/A

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

   2. associate-/l/ N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 31.6%

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

7. Taylor expanded in a around 0

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

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

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

   2. *-lowering-*.f64 99.4

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 90.9% 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 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 96.6%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

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

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

   5. associate-/l* N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. /-lowering-/.f64 91.0

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

8. Simplified 91.0%

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

Alternative 5: 90.9% 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 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 91.0

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

5. Simplified 91.0%

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

6. Final simplification 91.0%

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

Alternative 6: 81.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 81.6

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

5. Simplified 81.6%

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

6. Final simplification 81.6%

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

Reproduce

herbie shell --seed 2024198 
(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)))