Quadratic roots, narrow range

Percentage Accurate: 55.1% → 99.4%

Time: 11.2s

Alternatives: 11

Speedup: 3.6×

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: 55.1% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

4. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

6. Applied rewrites 56.0%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

    1. lift-*.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-neg N/A

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

    4. distribute-lft-in N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. lower-neg.f64 99.4

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

    14. lift-fma.f64 N/A

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

    15. *-commutative N/A

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

    16. lower-fma.f64 99.4

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.02)
   (/ (- (sqrt (fma b b (* (* c -4.0) a))) b) (* 2.0 a))
   (/ 0.5 (/ (fma (* (/ c b) a) 0.5 (* -0.5 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.02) {
		tmp = (sqrt(fma(b, b, ((c * -4.0) * a))) - b) / (2.0 * a);
	} else {
		tmp = 0.5 / (fma(((c / b) * a), 0.5, (-0.5 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(2.0 * a)) <= -0.02)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -4.0) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 / Float64(fma(Float64(Float64(c / b) * a), 0.5, Float64(-0.5 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 0.5 + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0200000000000000004

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 79.3

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

   4. Applied rewrites 79.3%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.9%

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

   4. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 89.8

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

   7. Applied rewrites 89.8%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 0.5, -0.5 \cdot b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 0.5, -0.5 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.02)
   (/ (- (sqrt (fma b b (* (* c -4.0) a))) b) (* 2.0 a))
   (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.02) {
		tmp = (sqrt(fma(b, b, ((c * -4.0) * a))) - b) / (2.0 * a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(2.0 * a)) <= -0.02)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -4.0) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0200000000000000004

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 79.3

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

   4. Applied rewrites 79.3%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.9%

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

   4. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

      \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.02)
   (* (- (sqrt (fma b b (* (* c -4.0) a))) b) (/ 0.5 a))
   (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.02) {
		tmp = (sqrt(fma(b, b, ((c * -4.0) * a))) - b) * (0.5 / a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(2.0 * a)) <= -0.02)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -4.0) * a))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0200000000000000004

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 79.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.1

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

   4. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 79.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.3

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

   6. Applied rewrites 79.3%

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

   if -0.0200000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.9%

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

   4. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

      \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.02:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

4. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

6. Applied rewrites 56.0%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 6: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

4. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

6. Applied rewrites 56.0%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. associate-*l* N/A

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

    9. lift-*.f64 N/A

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

    10. lower-*.f64 99.2

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

    11. lift-*.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 99.2

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

11. Applied rewrites 99.2%

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

12. Final simplification 99.2%

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

Alternative 7: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

4. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

6. Applied rewrites 56.0%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.2

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

9. Applied rewrites 99.2%

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

    1. lift-/.f64 N/A

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

    2. clear-num N/A

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

    3. associate-/r/ N/A

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

    4. lower-*.f64 N/A

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

11. Applied rewrites 99.2%

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

12. Final simplification 99.2%

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

Alternative 8: 82.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

4. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 82.2

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

7. Applied rewrites 82.2%

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

Alternative 9: 81.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 81.6

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

5. Applied rewrites 81.6%

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

6. Final simplification 81.6%

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

Alternative 10: 81.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot -2\right) \cdot a, \frac{c}{{b}^{5}}, \frac{-a}{\left(b \cdot b\right) \cdot b}\right), c, \frac{-1}{b}\right) \cdot c} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 81.4%

   \[\leadsto \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right)}{b} \cdot c \]
9. Add Preprocessing

Alternative 11: 64.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 64.9

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

5. Applied rewrites 64.9%

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

Reproduce

herbie shell --seed 2024240 
(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)))