Quadratic roots, wide range

Percentage Accurate: 17.6% → 97.5%

Time: 9.8s

Alternatives: 8

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: 17.6% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}, -2, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right), -0.25 \cdot \frac{20 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6} \cdot a}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* (* c c) (* (* a a) c)) (pow b 4.0))
   -2.0
   (fma
    -1.0
    (fma a (/ (* c c) (* b b)) c)
    (* -0.25 (/ (* 20.0 (* (pow a 4.0) (pow c 4.0))) (* (pow b 6.0) a)))))
  b))

C

double code(double a, double b, double c) {
	return fma((((c * c) * ((a * a) * c)) / pow(b, 4.0)), -2.0, fma(-1.0, fma(a, ((c * c) / (b * b)), c), (-0.25 * ((20.0 * (pow(a, 4.0) * pow(c, 4.0))) / (pow(b, 6.0) * a))))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64(Float64(c * c) * Float64(Float64(a * a) * c)) / (b ^ 4.0)), -2.0, fma(-1.0, fma(a, Float64(Float64(c * c) / Float64(b * b)), c), Float64(-0.25 * Float64(Float64(20.0 * Float64((a ^ 4.0) * (c ^ 4.0))) / Float64((b ^ 6.0) * a))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-1.0 * N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] + N[(-0.25 * N[(N[(20.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 22.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(20 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}, \frac{-1}{4}, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]

9. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

11. Applied rewrites 97.0%

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

13. Applied rewrites 97.0%

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

14. Final simplification 97.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}, -2, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right), -0.25 \cdot \frac{20 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6} \cdot a}\right)\right)}{b} \]
15. Add Preprocessing

Alternative 2: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot a\right) \cdot 20}{{b}^{7}}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma
   (fma
    (fma
     (/ (* (* (* (* a a) c) a) 20.0) (pow b 7.0))
     -0.25
     (/ (* -2.0 (* a a)) (pow b 5.0)))
    c
    (/ (- a) (pow b 3.0)))
   c
   (/ -1.0 b))
  c))

C

double code(double a, double b, double c) {
	return fma(fma(fma((((((a * a) * c) * a) * 20.0) / pow(b, 7.0)), -0.25, ((-2.0 * (a * a)) / pow(b, 5.0))), c, (-a / pow(b, 3.0))), c, (-1.0 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(fma(fma(Float64(Float64(Float64(Float64(Float64(a * a) * c) * a) * 20.0) / (b ^ 7.0)), -0.25, Float64(Float64(-2.0 * Float64(a * a)) / (b ^ 5.0))), 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[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -0.25 + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $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 22.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(20 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}, \frac{-1}{4}, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]
9. Step-by-step derivation

10. Applied rewrites 96.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot a\right)}{{b}^{7}}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]

11. Final simplification 96.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot a\right) \cdot 20}{{b}^{7}}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
12. Add Preprocessing

Alternative 3: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{4}}, -2, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -1, -c\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (/ (* (pow c 3.0) (* a a)) (pow b 4.0))
   -2.0
   (fma (/ (* (* c c) a) (* b b)) -1.0 (- c)))
  b))

C

double code(double a, double b, double c) {
	return fma(((pow(c, 3.0) * (a * a)) / pow(b, 4.0)), -2.0, fma((((c * c) * a) / (b * b)), -1.0, -c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 4.0)), -2.0, fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -1.0, Float64(-c))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -1.0 + (-c)), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 22.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 95.9%

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

6. Final simplification 95.9%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{4}}, -2, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -1, -c\right)\right)}{b} \]
7. Add Preprocessing

Alternative 4: 96.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 95.9%

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

9. Final simplification 95.9%

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

Alternative 5: 96.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.6%

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

7. Applied rewrites 95.6%

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

8. Final simplification 95.6%

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

Alternative 6: 95.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 94.0

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

5. Applied rewrites 94.0%

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

Alternative 7: 95.2% 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 22.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(20 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}, \frac{-1}{4}, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), c, -\frac{a}{{b}^{3}}\right), c, -\frac{1}{b}\right) \cdot c \]

9. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
10. 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 94.0

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

11. Applied rewrites 94.0%

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

12. Final simplification 94.0%

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

Alternative 8: 90.5% 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 22.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 87.6

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

5. Applied rewrites 87.6%

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

Reproduce

herbie shell --seed 2024312 
(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)))