Quadratic roots, narrow range

Percentage Accurate: 55.8% → 99.6%

Time: 14.0s

Alternatives: 7

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: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in a around inf

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 57.3

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

5. Simplified 57.3%

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

7. Applied egg-rr 58.9%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.4

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

10. Simplified 99.4%

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

    1. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. associate-*l* N/A

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

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

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

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

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

    12. *-lowering-*.f64 99.5

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

12. Applied egg-rr 99.5%

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

Alternative 2: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.7:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.7)
   (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* a -4.0))))))
   (* (* c 2.0) (/ 1.0 (* 2.0 (fma a (/ c b) (- b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.7) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = (c * 2.0) * (1.0 / (2.0 * fma(a, (c / b), -b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.7)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(c * 2.0) * Float64(1.0 / Float64(2.0 * fma(a, Float64(c / b), Float64(-b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.7], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] * N[(1.0 / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $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)) < -0.69999999999999996

   1. Initial program 82.3%

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

   4. Applied egg-rr 82.8%

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

   if -0.69999999999999996 < (/.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 51.7%

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

   3. Taylor expanded in a around inf

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 51.4

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

   5. Simplified 51.4%

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

   7. Applied egg-rr 53.0%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.4

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

   10. Simplified 99.4%

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

   11. Taylor expanded in a around 0

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

       1. distribute-lft-out-- N/A

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

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

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

       3. sub-neg N/A

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

       4. associate-/l* N/A

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

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

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

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

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

       7. neg-lowering-neg.f64 87.2

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

   13. Simplified 87.2%

       \[\leadsto \left(c \cdot 2\right) \cdot \frac{1}{\color{blue}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.7:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in a around inf

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 57.3

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

5. Simplified 57.3%

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

7. Applied egg-rr 58.9%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.4

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

10. Simplified 99.4%

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

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

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

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

    4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

    9. associate-*r* N/A

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

    10. *-commutative N/A

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

    11. associate-*l* N/A

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

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

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

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

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

    14. *-lowering-*.f64 99.4

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 4: 82.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (* (* c 2.0) (/ 1.0 (* 2.0 (fma a (/ c b) (- b))))))

C

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

Julia

function code(a, b, c)
	return Float64(Float64(c * 2.0) * Float64(1.0 / Float64(2.0 * fma(a, Float64(c / b), Float64(-b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * 2.0), $MachinePrecision] * N[(1.0 / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in a around inf

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 57.3

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

5. Simplified 57.3%

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

7. Applied egg-rr 58.9%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.4

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

10. Simplified 99.4%

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

11. Taylor expanded in a around 0

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

    1. distribute-lft-out-- N/A

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

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

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

    3. sub-neg N/A

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

    4. associate-/l* N/A

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

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

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

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

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

    7. neg-lowering-neg.f64 82.1

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

13. Simplified 82.1%

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

Alternative 5: 81.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{2} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{2 \cdot a} \]

5. Simplified 91.7%

   \[\leadsto \frac{\color{blue}{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.5, \frac{{c}^{4} \cdot 20}{{b}^{6}} \cdot \frac{a}{b}, \frac{-4 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{-2 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\right), -2 \cdot \frac{c}{b}\right)}}{2 \cdot a} \]

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

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

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

   6. associate-/l* N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

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

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

   16. /-lowering-/.f64 81.5

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

8. Simplified 81.5%

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

Alternative 6: 81.4% 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 57.6%

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

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 81.5

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

5. Simplified 81.5%

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

6. Final simplification 81.5%

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

Alternative 7: 64.1% 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 57.6%

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

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

   4. neg-lowering-neg.f64 63.1

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

5. Simplified 63.1%

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

Reproduce

herbie shell --seed 2024198 
(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)))