Quadratic roots, wide range

Percentage Accurate: 17.1% → 97.9%

Time: 11.7s

Alternatives: 6

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: 17.1% 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: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -0.25, \frac{20 \cdot {c}^{4}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (fma
   a
   (fma
    (* a -0.25)
    (/ (* 20.0 (pow c 4.0)) (pow b 7.0))
    (/ (* -2.0 (* c (* c c))) (pow b 5.0)))
   (/ (* c c) (* b (* b (- b)))))
  (/ c (- b))))

C

double code(double a, double b, double c) {
	return fma(a, fma(a, fma((a * -0.25), ((20.0 * pow(c, 4.0)) / pow(b, 7.0)), ((-2.0 * (c * (c * c))) / pow(b, 5.0))), ((c * c) / (b * (b * -b)))), (c / -b));
}

Julia

function code(a, b, c)
	return fma(a, fma(a, fma(Float64(a * -0.25), Float64(Float64(20.0 * (c ^ 4.0)) / (b ^ 7.0)), Float64(Float64(-2.0 * Float64(c * Float64(c * c))) / (b ^ 5.0))), Float64(Float64(c * c) / Float64(b * Float64(b * Float64(-b))))), Float64(c / Float64(-b)))
end

Wolfram

code[a_, b_, c_] := N[(a * N[(a * N[(N[(a * -0.25), $MachinePrecision] * N[(N[(20.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in b around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 19.2

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

5. Simplified 19.2%

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

6. Taylor expanded in a around 0

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

8. Simplified 97.8%

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

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{4} \cdot a, \color{blue}{20 \cdot \frac{{c}^{4}}{{b}^{7}}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{4} \cdot a, \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{7}}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    2. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{4} \cdot a, \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{7}}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{4} \cdot a, \frac{\color{blue}{20 \cdot {c}^{4}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    4. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{4} \cdot a, \frac{20 \cdot \color{blue}{{c}^{4}}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}\right), \frac{c}{\mathsf{neg}\left(b\right)}\right) \]

    5. lower-pow.f64 97.8

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.25 \cdot a, \frac{20 \cdot {c}^{4}}{\color{blue}{{b}^{7}}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right) \]

11. Simplified 97.8%

    \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.25 \cdot a, \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{7}}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right) \]

12. Final simplification 97.8%

    \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -0.25, \frac{20 \cdot {c}^{4}}{{b}^{7}}, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right) \]
13. Add Preprocessing

Alternative 2: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right), -2 \cdot \left(a \cdot a\right), {c}^{4} \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -5\right)\right)}{{b}^{7}} - \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (/
   (fma
    (* (* c (* c c)) (* b b))
    (* -2.0 (* a a))
    (* (pow c 4.0) (* (* a (* a a)) -5.0)))
   (pow b 7.0))
  (/ (fma a (/ (* c c) (* b b)) c) b)))

C

double code(double a, double b, double c) {
	return (fma(((c * (c * c)) * (b * b)), (-2.0 * (a * a)), (pow(c, 4.0) * ((a * (a * a)) * -5.0))) / pow(b, 7.0)) - (fma(a, ((c * c) / (b * b)), c) / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(Float64(c * Float64(c * c)) * Float64(b * b)), Float64(-2.0 * Float64(a * a)), Float64((c ^ 4.0) * Float64(Float64(a * Float64(a * a)) * -5.0))) / (b ^ 7.0)) - Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in a around 0

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

5. Simplified 97.7%

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

6. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\right)\right) + \color{blue}{\frac{-5 \cdot \left({a}^{3} \cdot {c}^{4}\right) + -2 \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {c}^{3}\right)\right)}{{b}^{7}}} \]

8. Simplified 97.7%

   \[\leadsto \left(-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\right) + \color{blue}{\frac{\mathsf{fma}\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right), \left(a \cdot a\right) \cdot -2, {c}^{4} \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -5\right)\right)}{{b}^{7}}} \]

9. Final simplification 97.7%

   \[\leadsto \frac{\mathsf{fma}\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right), -2 \cdot \left(a \cdot a\right), {c}^{4} \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -5\right)\right)}{{b}^{7}} - \frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b} \]
10. Add Preprocessing

Alternative 3: 97.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in 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. Simplified 96.9%

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

6. Final simplification 96.9%

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

Alternative 4: 95.7% 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 19.2%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. 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-/l* 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

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 94.8

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

5. Simplified 94.8%

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

6. Final simplification 94.8%

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

Alternative 5: 95.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in c around 0

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. cube-mult N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-/.f64 94.4

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

5. Simplified 94.4%

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

6. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

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

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

   5. distribute-rgt-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-*.f64 N/A

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

8. Simplified 94.4%

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

9. Taylor expanded in c around 0

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

11. Simplified 94.4%

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

Alternative 6: 90.9% 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(c / Float64(-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 19.2%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 89.2

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

5. Simplified 89.2%

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

6. Final simplification 89.2%

   \[\leadsto \frac{c}{-b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))