Quadratic roots, wide range

Percentage Accurate: 18.3% → 99.9%

Time: 14.3s

Alternatives: 5

Speedup: 3.3×

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: 18.3% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

4. Applied egg-rr 17.0%

   \[\leadsto \frac{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-1}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)}}{2 \cdot a} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

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

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

6. Applied egg-rr 17.4%

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

7. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot c\right)} \cdot \frac{\frac{-1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.5

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

9. Simplified 99.5%

   \[\leadsto \color{blue}{\left(c \cdot 4\right)} \cdot \frac{\frac{-1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2} \]
10. Step-by-step derivation

    1. clear-num N/A

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

    2. un-div-inv N/A

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

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

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

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

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

    5. associate-/r/ N/A

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

    6. metadata-eval N/A

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

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

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

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

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

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

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

    10. +-commutative N/A

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

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

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

    12. associate-*r* N/A

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

    13. *-commutative N/A

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

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

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

    15. *-lowering-*.f64 99.9

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

11. Applied egg-rr 99.9%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(c \cdot 4\right) \cdot \frac{-0.5}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (* c 4.0) (/ -0.5 (+ b (sqrt (fma b b (* -4.0 (* c a))))))))

C

double code(double a, double b, double c) {
	return (c * 4.0) * (-0.5 / (b + sqrt(fma(b, b, (-4.0 * (c * a))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(c * 4.0) * Float64(-0.5 / Float64(b + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * 4.0), $MachinePrecision] * N[(-0.5 / N[(b + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

4. Applied egg-rr 17.0%

   \[\leadsto \frac{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-1}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)}}{2 \cdot a} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

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

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

6. Applied egg-rr 17.4%

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

7. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot c\right)} \cdot \frac{\frac{-1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.5

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

9. Simplified 99.5%

   \[\leadsto \color{blue}{\left(c \cdot 4\right)} \cdot \frac{\frac{-1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

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

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

    3. associate-/l/ N/A

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

    4. associate-/r* N/A

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

    5. metadata-eval N/A

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

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

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

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

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

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

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

    9. +-commutative N/A

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

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

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

    11. associate-*r* N/A

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

    12. *-commutative N/A

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

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

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

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

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

    15. *-lowering-*.f64 99.5

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

    \[\leadsto \left(c \cdot 4\right) \cdot \frac{-0.5}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \]
13. Add Preprocessing

Alternative 3: 95.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. +-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 17.0

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

4. Applied egg-rr 17.0%

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

   1. clear-num N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. rem-square-sqrt N/A

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

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

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

   11. rem-square-sqrt N/A

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

   12. +-commutative N/A

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

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

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

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

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

   15. *-lowering-*.f64 16.9

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

6. Applied egg-rr 16.9%

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

7. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 94.8

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

9. Simplified 94.8%

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

    1. clear-num N/A

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

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

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

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

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

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

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

    5. *-lowering-*.f64 95.1

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

11. Applied egg-rr 95.1%

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

Alternative 4: 90.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

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

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   4. /-lowering-/.f64 90.8

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

5. Simplified 90.8%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

   3. /-lowering-/.f64 90.8

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

7. Applied egg-rr 90.8%

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

8. Final simplification 90.8%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Alternative 5: 1.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. +-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 17.0

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

4. Applied egg-rr 17.0%

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

   1. clear-num N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. rem-square-sqrt N/A

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

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

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

   11. rem-square-sqrt N/A

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

   12. +-commutative N/A

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

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

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

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

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

   15. *-lowering-*.f64 16.9

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

6. Applied egg-rr 16.9%

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

7. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 94.8

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

9. Simplified 94.8%

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

10. Taylor expanded in c around inf

    \[\leadsto \color{blue}{\frac{b}{a}} \]
11. Step-by-step derivation

    1. /-lowering-/.f64 1.6

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

12. Simplified 1.6%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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)))