Quadratic roots, narrow range

Percentage Accurate: 55.7% → 99.5%

Time: 10.2s

Alternatives: 6

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: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied rewrites 55.9%

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

6. Applied rewrites 57.6%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in c around inf

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

    1. lower-*.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(a, \frac{c}{b}, -b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.14)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (* 2.0 a))
   (/ (* 2.0 c) (* (fma a (/ c b) (- b)) 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.14) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (fma(a, (c / b), -b) * 2.0);
	}
	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)) <= -0.14)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(fma(a, Float64(c / b), Float64(-b)) * 2.0));
	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], -0.14], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] * 2.0), $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)) < -0.14000000000000001

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 84.5

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 84.6

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

   4. Applied rewrites 84.6%

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

   if -0.14000000000000001 < (/.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 49.0%

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in a around 0

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

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

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

       2. lower-*.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-neg.f64 87.5

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

   12. Applied rewrites 87.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.14:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(a, \frac{c}{b}, -b\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.14:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(a, \frac{c}{b}, -b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.14)
   (* (/ 0.5 a) (- (sqrt (fma (* c -4.0) a (* b b))) b))
   (/ (* 2.0 c) (* (fma a (/ c b) (- b)) 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.14) {
		tmp = (0.5 / a) * (sqrt(fma((c * -4.0), a, (b * b))) - b);
	} else {
		tmp = (2.0 * c) / (fma(a, (c / b), -b) * 2.0);
	}
	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)) <= -0.14)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(fma(a, Float64(c / b), Float64(-b)) * 2.0));
	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], -0.14], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] * 2.0), $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)) < -0.14000000000000001

   1. Initial program 84.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 84.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      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 84.5

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

   4. Applied rewrites 84.5%

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

   if -0.14000000000000001 < (/.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 49.0%

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

   4. Applied rewrites 49.1%

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in a around 0

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

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

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

       2. lower-*.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-neg.f64 87.5

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

   12. Applied rewrites 87.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.14:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(a, \frac{c}{b}, -b\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied rewrites 55.9%

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

6. Applied rewrites 57.6%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. *-commutative N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 5: 81.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

4. Applied rewrites 55.9%

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

6. Applied rewrites 57.6%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in a around 0

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

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

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

    2. lower-*.f64 N/A

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

    3. sub-neg N/A

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

    4. associate-/l* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-neg.f64 81.4

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

12. Applied rewrites 81.4%

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

13. Final simplification 81.4%

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

Alternative 6: 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(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 55.8%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 64.0

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

5. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2024331 
(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)))