Quadratic roots, wide range

Percentage Accurate: 17.9% → 97.5%

Time: 11.5s

Alternatives: 9

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.9% 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: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}}, 0.5, \frac{\left(c \cdot c\right) \cdot a}{{b}^{5}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (fma
   (fma (fma (/ c (pow b 3.0)) 0.5 (/ (* (* c c) a) (pow b 5.0))) a (/ 0.5 b))
   a
   (* (/ b c) -0.5))))

C

double code(double a, double b, double c) {
	return 0.5 / fma(fma(fma((c / pow(b, 3.0)), 0.5, (((c * c) * a) / pow(b, 5.0))), a, (0.5 / b)), a, ((b / c) * -0.5));
}

Julia

function code(a, b, c)
	return Float64(0.5 / fma(fma(fma(Float64(c / (b ^ 3.0)), 0.5, Float64(Float64(Float64(c * c) * a) / (b ^ 5.0))), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.5)))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 18.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 18.4

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

4. Applied rewrites 18.4%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-*.f64 N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 18.7%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

9. Applied rewrites 97.2%

   \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}}, 0.5, \frac{\left(c \cdot c\right) \cdot a}{{b}^{5}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}} \]
10. Add Preprocessing

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(c \cdot a, -5, \left(b \cdot b\right) \cdot -2\right) \cdot \left(a \cdot a\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma
   (fma
    (/ (* (fma (* c a) -5.0 (* (* b b) -2.0)) (* a a)) (pow b 7.0))
    c
    (/ (- a) (pow b 3.0)))
   c
   (/ -1.0 b))
  c))

C

double code(double a, double b, double c) {
	return fma(fma(((fma((c * a), -5.0, ((b * b) * -2.0)) * (a * a)) / pow(b, 7.0)), c, (-a / pow(b, 3.0))), c, (-1.0 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(fma(Float64(Float64(fma(Float64(c * a), -5.0, Float64(Float64(b * b) * -2.0)) * Float64(a * a)) / (b ^ 7.0)), c, Float64(Float64(-a) / (b ^ 3.0))), c, Float64(-1.0 / b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -5.0 + N[(N[(b * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * c + N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 97.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, c, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{a}^{2} \cdot \left(-5 \cdot \left(a \cdot c\right) + -2 \cdot {b}^{2}\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
10. Step-by-step derivation

11. Applied rewrites 97.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(c \cdot a, -5, \left(b \cdot b\right) \cdot -2\right) \cdot \left(a \cdot a\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
12. Add Preprocessing

Alternative 3: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (/ (* (* (* (pow c 3.0) a) a) -2.0) (pow b 4.0))
   (fma (/ c b) (/ (* c a) b) c))
  b))

C

double code(double a, double b, double c) {
	return (((((pow(c, 3.0) * a) * a) * -2.0) / pow(b, 4.0)) - fma((c / b), ((c * a) / b), c)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64((c ^ 3.0) * a) * a) * -2.0) / (b ^ 4.0)) - fma(Float64(c / b), Float64(Float64(c * a) / b), c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 96.8%

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

Alternative 4: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (fma (- (/ (* -2.0 (* (* a a) c)) (pow b 4.0)) (/ a (* b b))) c -1.0) c)
  b))

C

double code(double a, double b, double c) {
	return (fma((((-2.0 * ((a * a) * c)) / pow(b, 4.0)) - (a / (b * b))), c, -1.0) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * c)) / (b ^ 4.0)) - Float64(a / Float64(b * b))), c, -1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 96.8%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 96.7%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \]
12. Add Preprocessing

Alternative 5: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \frac{c}{{b}^{3}}, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.5 (fma (fma (* 0.5 (/ c (pow b 3.0))) a (/ 0.5 b)) a (* (/ b c) -0.5))))

C

double code(double a, double b, double c) {
	return 0.5 / fma(fma((0.5 * (c / pow(b, 3.0))), a, (0.5 / b)), a, ((b / c) * -0.5));
}

Julia

function code(a, b, c)
	return Float64(0.5 / fma(fma(Float64(0.5 * Float64(c / (b ^ 3.0))), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.5)))
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(N[(0.5 * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 18.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 18.4

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

4. Applied rewrites 18.4%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 96.5%

   \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \frac{c}{{b}^{3}}, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}} \]
8. Add Preprocessing

Alternative 6: 95.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in a around 0

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

   1. distribute-lft-out N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-/.f64 95.2

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

8. Applied rewrites 95.2%

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

Alternative 7: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. associate-*r/ N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-/l* N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 95.1%

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

7. Applied rewrites 95.1%

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

8. Final simplification 95.1%

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

Alternative 8: 95.3% 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 18.4%

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

3. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. associate-*r/ N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-/l* N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 95.1%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 95.1%

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

9. Final simplification 95.1%

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

Alternative 9: 90.3% 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 18.4%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 90.2

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

5. Applied rewrites 90.2%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))