Quadratic roots, narrow range

Percentage Accurate: 56.1% → 99.6%

Time: 10.6s

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: 56.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

4. Applied rewrites 53.3%

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

6. Applied rewrites 55.7%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.08)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (* 2.0 a))
   (/ (- (- c) (/ (* (* c c) a) (* b b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.08) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = (-c - (((c * c) * a) / (b * b))) / b;
	}
	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)) <= -0.08)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(Float64(c * c) * a) / Float64(b * b))) / b);
	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], -0.08], 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[((-c) - N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0800000000000000017

   1. Initial program 78.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 78.7

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

      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 78.7

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

   4. Applied rewrites 78.7%

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

   if -0.0800000000000000017 < (/.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 45.6%

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in b around inf

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

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -0.08)
   (* (/ 0.5 a) (- (sqrt (fma (* c -4.0) a (* b b))) b))
   (/ (- (- c) (/ (* (* c c) a) (* b b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -0.08) {
		tmp = (0.5 / a) * (sqrt(fma((c * -4.0), a, (b * b))) - b);
	} else {
		tmp = (-c - (((c * c) * a) / (b * b))) / b;
	}
	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)) <= -0.08)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(Float64(c * c) * a) / Float64(b * b))) / b);
	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], -0.08], 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[((-c) - N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0800000000000000017

   1. Initial program 78.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 78.6

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

      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 78.6

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

   4. Applied rewrites 78.6%

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

   if -0.0800000000000000017 < (/.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 45.6%

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in b around inf

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

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -0.08:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

4. Applied rewrites 53.3%

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

5. Taylor expanded in b around inf

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

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 82.8

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

7. Applied rewrites 82.8%

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

8. Final simplification 82.8%

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

Alternative 5: 81.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -c * ((((a * c) / (b * b)) + 1.0) / 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 * ((((a * c) / (b * b)) + 1.0d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-/.f64 82.7

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

5. Applied rewrites 82.7%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 65.9%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 82.6%

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

12. Final simplification 82.6%

    \[\leadsto \left(-c\right) \cdot \frac{\frac{a \cdot c}{b \cdot b} + 1}{b} \]
13. Add Preprocessing

Alternative 6: 63.9% 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 53.7%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 65.9

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

5. Applied rewrites 65.9%

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

Reproduce

herbie shell --seed 2024268 
(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)))