Quadratic roots, wide range

Percentage Accurate: 17.9% → 99.5%

Time: 13.5s

Alternatives: 7

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.9% 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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.6%

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

4. Applied rewrites 15.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-div N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 16.1%

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

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. rem-square-sqrt N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/r/ N/A

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

    5. lift-/.f64 N/A

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

    6. associate-*r/ N/A

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

    7. lower-/.f64 N/A

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

11. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.6%

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

4. Applied rewrites 15.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-div N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 16.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 16.0

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

8. Applied rewrites 16.0%

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

9. Taylor expanded in b around 0

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

    1. lower-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 99.5

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. clear-num N/A

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

   10. flip-- N/A

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

   11. lift--.f64 N/A

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

4. Applied rewrites 16.0%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 95.1

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

7. Applied rewrites 95.1%

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

8. Final simplification 95.1%

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

Alternative 4: 95.4% 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 15.6%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-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. lower-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. lower-*.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. lower-/.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. lower-*.f64 95.1

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

5. Applied rewrites 95.1%

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

6. Final simplification 95.1%

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

Alternative 5: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a / b) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.6%

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

4. Applied rewrites 15.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-div N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 16.1%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 94.9

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

9. Applied rewrites 94.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
10. Add Preprocessing

Alternative 6: 90.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. lower-/.f64 N/A

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

   4. lower-neg.f64 91.6

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

5. Applied rewrites 91.6%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 7: 3.3% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 15.6%

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

4. Applied rewrites 15.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

6. Applied rewrites 16.2%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.3

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

9. Applied rewrites 3.3%

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

Reproduce

herbie shell --seed 2024226 
(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)))