Quadratic roots, narrow range

Percentage Accurate: 55.5% → 99.3%

Time: 11.2s

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: 55.5% 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: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 56.0

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

5. Applied rewrites 56.0%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 56.8%

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

8. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 2: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.45:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.45)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/ (fma -1.0 (/ (* a (* c c)) (* b b)) (- c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.45) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = fma(-1.0, ((a * (c * c)) / (b * b)), -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.45)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(-1.0, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.45], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $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)) < -0.450000000000000011

   1. Initial program 85.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 85.7

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

   4. Applied rewrites 85.7%

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

   if -0.450000000000000011 < (/.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 50.6%

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.5%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-c, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)}{a}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \frac{-a}{{b}^{3}}}{b \cdot b}, \frac{a}{{b}^{3}}\right), \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. 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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 85.7

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

   10. Applied rewrites 85.7%

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

Alternative 3: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.45:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.45)
   (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (fma -1.0 (/ (* a (* c c)) (* b b)) (- c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.45) {
		tmp = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else {
		tmp = fma(-1.0, ((a * (c * c)) / (b * b)), -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.45)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(fma(-1.0, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.45], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $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)) < -0.450000000000000011

   1. Initial program 85.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 85.6

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

      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 85.6

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

   4. Applied rewrites 85.6%

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

   if -0.450000000000000011 < (/.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 50.6%

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.5%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-c, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)}{a}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \frac{-a}{{b}^{3}}}{b \cdot b}, \frac{a}{{b}^{3}}\right), \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. 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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 85.7

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

   10. Applied rewrites 85.7%

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

Alternative 4: 81.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

4. Applied rewrites 56.2%

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

5. Taylor expanded in c around 0

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

7. Applied rewrites 90.3%

   \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-c, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)}{a}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \frac{-a}{{b}^{3}}}{b \cdot b}, \frac{a}{{b}^{3}}\right), \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}} \]

8. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
9. 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. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 80.7

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

10. Applied rewrites 80.7%

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

Alternative 5: 64.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 56.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 63.8

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

5. Applied rewrites 63.8%

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

Alternative 6: 1.6% accurate, 4.2× 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 / 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 56.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 63.8

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

5. Applied rewrites 63.8%

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

7. Applied rewrites 63.8%

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

9. Applied rewrites 63.7%

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

11. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024296 
(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)))