Quadratic roots, narrow range

Percentage Accurate: 55.9% → 99.6%

Time: 13.7s

Alternatives: 8

Speedup: 2.3×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.9% 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: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 99.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 99.6

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.6

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

9. Applied egg-rr 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

9. Applied egg-rr 99.4%

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

Alternative 4: 64.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 99.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 99.6

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.6

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

9. Applied egg-rr 99.6%

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

10. Taylor expanded in c around -inf

    \[\leadsto \frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-1 \cdot c\right)}\right)}} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. lower-neg.f64 64.2

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

12. Simplified 64.2%

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

13. Final simplification 64.2%

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

Alternative 5: 64.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

8. Taylor expanded in c around -inf

   \[\leadsto \frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(a, \color{blue}{-1 \cdot c}, b \cdot b\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 64.2

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

10. Simplified 64.2%

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

Alternative 6: 63.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{-2 \cdot c}{b \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-2.0 * c) / (b * 2.0);
}

Python

def code(a, b, c):
	return (-2.0 * c) / (b * 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

8. Taylor expanded in a around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 12.6

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

10. Simplified 12.6%

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

11. Taylor expanded in b around 0

    \[\leadsto \frac{-2 \cdot c}{\color{blue}{2 \cdot b}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{-2 \cdot c}{\color{blue}{b \cdot 2}} \]

    2. lower-*.f64 63.7

       \[\leadsto \frac{-2 \cdot c}{\color{blue}{b \cdot 2}} \]

13. Simplified 63.7%

    \[\leadsto \frac{-2 \cdot c}{\color{blue}{b \cdot 2}} \]
14. Add Preprocessing

Alternative 7: 18.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-2 \cdot c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (-2.0 * 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 = ((-2.0d0) * c) / b
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * c) / b;
}

Python

def code(a, b, c):
	return (-2.0 * c) / b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 11.8%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 99.6

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

7. Simplified 99.6%

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

8. Taylor expanded in a around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 12.6

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

10. Simplified 12.6%

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

11. Taylor expanded in b around 0

    \[\leadsto \color{blue}{-2 \cdot \frac{c}{b}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{-2 \cdot c}{b}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{-2 \cdot c}{b}} \]

    3. lower-*.f64 18.8

       \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b} \]

13. Simplified 18.8%

    \[\leadsto \color{blue}{\frac{-2 \cdot c}{b}} \]
14. Add Preprocessing

Alternative 8: 11.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.5 * (b / 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 = (-0.5d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (b / a);
}

Python

def code(a, b, c):
	return -0.5 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

4. Applied egg-rr 2.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{\mathsf{fma}\left(-64, \left(a \cdot c\right) \cdot \left(a \cdot \left(c \cdot \left(a \cdot c\right)\right)\right), b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right), \left(16 \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

5. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 11.9

      \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

7. Simplified 11.9%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))