Quadratic roots, narrow range

Percentage Accurate: 55.7% → 91.1%

Time: 10.3s

Alternatives: 8

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: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-c\right) - \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left({a}^{3} \cdot 5, c, \left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2\right)\right)}{{b}^{6}}, c, \frac{a}{b \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (- c)
   (*
    (* c c)
    (fma
     (/ (fma (* (pow a 3.0) 5.0) c (* (* b b) (* (* a a) 2.0))) (pow b 6.0))
     c
     (/ a (* b b)))))
  b))

C

double code(double a, double b, double c) {
	return (-c - ((c * c) * fma((fma((pow(a, 3.0) * 5.0), c, ((b * b) * ((a * a) * 2.0))) / pow(b, 6.0)), c, (a / (b * b))))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) - Float64(Float64(c * c) * fma(Float64(fma(Float64((a ^ 3.0) * 5.0), c, Float64(Float64(b * b) * Float64(Float64(a * a) * 2.0))) / (b ^ 6.0)), c, Float64(a / Float64(b * b))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) - N[(N[(c * c), $MachinePrecision] * N[(N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * 5.0), $MachinePrecision] * c + N[(N[(b * b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * c + N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 93.8%

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

6. Taylor expanded in b around -inf

   \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

8. Applied rewrites 93.9%

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

9. Taylor expanded in c around 0

   \[\leadsto \frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{-b} \]
10. Step-by-step derivation

11. Applied rewrites 93.9%

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

12. Taylor expanded in b around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + 5 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}, c, \frac{a}{b \cdot b}\right) \cdot \left(c \cdot c\right) + c}{-b} \]
13. Step-by-step derivation

14. Applied rewrites 93.9%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(5 \cdot {a}^{3}, c, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{6}}, c, \frac{a}{b \cdot b}\right) \cdot \left(c \cdot c\right) + c}{-b} \]

15. Final simplification 93.9%

    \[\leadsto \frac{\left(-c\right) - \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left({a}^{3} \cdot 5, c, \left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2\right)\right)}{{b}^{6}}, c, \frac{a}{b \cdot b}\right)}{b} \]
16. Add Preprocessing

Alternative 2: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\frac{\left(a \cdot a\right) \cdot c}{b \cdot b}}{b \cdot b}, 2, \frac{a}{b \cdot b}\right) \cdot c, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 7.4)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 2.0 a)))
     (/
      (fma
       c
       (* (fma (/ (/ (* (* a a) c) (* b b)) (* b b)) 2.0 (/ a (* b b))) c)
       c)
      (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (2.0 * a));
	} else {
		tmp = fma(c, (fma(((((a * a) * c) / (b * b)) / (b * b)), 2.0, (a / (b * b))) * c), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(c, Float64(fma(Float64(Float64(Float64(Float64(a * a) * c) / Float64(b * b)) / Float64(b * b)), 2.0, Float64(a / Float64(b * b))) * c), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   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

   4. Applied rewrites 84.5%

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

   6. Applied rewrites 85.9%

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

   if 7.4000000000000004 < b

   1. Initial program 48.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 93.7%

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

   13. Applied rewrites 93.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\frac{\left(a \cdot a\right) \cdot c}{b \cdot b}}{b \cdot b}, 2, \frac{a}{b \cdot b}\right) \cdot c, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 10.0)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 2.0 a)))
     (/ (fma a (/ (* c c) (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 10.0) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (2.0 * a));
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 10.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 10.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 10

   1. Initial program 84.0%

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.5%

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

   if 10 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in c around 0

      \[\leadsto \frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{-b} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.3%

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

   12. Taylor expanded in b around inf

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

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 88.0

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

   14. Applied rewrites 88.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.0)
   (/ (- (sqrt (fma b b (* (* -4.0 c) a))) b) (* 2.0 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.0) {
		tmp = (sqrt(fma(b, b, ((-4.0 * c) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10

   1. Initial program 84.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 84.2

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

   4. Applied rewrites 84.2%

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

   if 10 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in c around 0

      \[\leadsto \frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{-b} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.3%

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

   12. Taylor expanded in b around inf

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

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 88.0

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

   14. Applied rewrites 88.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.0)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.0) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.0], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10

   1. Initial program 84.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 84.1

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

      9. lift-+.f64 N/A

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

      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 84.1

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

   4. Applied rewrites 84.1%

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

   if 10 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in c around 0

      \[\leadsto \frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{-b} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.3%

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

   12. Taylor expanded in b around inf

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

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 88.0

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

   14. Applied rewrites 88.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 93.8%

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

6. Taylor expanded in b around -inf

   \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

8. Applied rewrites 93.9%

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

9. Taylor expanded in c around 0

   \[\leadsto \frac{{c}^{2} \cdot \left(c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{4}} + 5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right) + \frac{a}{{b}^{2}}\right) + c}{-b} \]
10. Step-by-step derivation

11. Applied rewrites 93.9%

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

12. Taylor expanded in b around inf

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

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

    7. associate-/l* N/A

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

    8. lower-fma.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 83.6

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

14. Applied rewrites 83.6%

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

15. Final simplification 83.6%

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

Alternative 7: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 93.8%

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

6. Taylor expanded in b around -inf

   \[\leadsto -1 \cdot \color{blue}{\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

8. Applied rewrites 93.9%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 90.5%

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

12. Taylor expanded in c around 0

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

14. Applied rewrites 83.5%

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

15. Final simplification 83.5%

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

Alternative 8: 64.2% 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 53.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 66.0

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

5. Applied rewrites 66.0%

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

Reproduce

herbie shell --seed 2024278 
(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)))