Quadratic roots, wide range

Percentage Accurate: 18.3% → 98.3%

Time: 38.4s

Alternatives: 11

Speedup: 29.0×

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: 18.3% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} + c \cdot \left(-42 \cdot \frac{c \cdot {a}^{3}}{{b}^{11}} + -14 \cdot \frac{{a}^{2}}{{b}^{9}}\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (fma
      -2.0
      (/ (pow c 3.0) (pow b 5.0))
      (*
       (pow c 4.0)
       (+
        (* -5.0 (/ a (pow b 7.0)))
        (*
         c
         (+
          (* -42.0 (/ (* c (pow a 3.0)) (pow b 11.0)))
          (* -14.0 (/ (pow a 2.0) (pow b 9.0)))))))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (pow(c, 4.0) * ((-5.0 * (a / pow(b, 7.0))) + (c * ((-42.0 * ((c * pow(a, 3.0)) / pow(b, 11.0))) + (-14.0 * (pow(a, 2.0) / pow(b, 9.0))))))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64(a / (b ^ 7.0))) + Float64(c * Float64(Float64(-42.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 11.0))) + Float64(-14.0 * Float64((a ^ 2.0) / (b ^ 9.0))))))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-42.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 11.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-14.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in a around 0 97.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.25 \cdot \frac{4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + a \cdot \left(-0.25 \cdot \frac{a \cdot \left(2 \cdot \frac{c \cdot \left(2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}} + \left(2 \cdot \frac{{c}^{2} \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{4}} + 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + -0.25 \cdot \frac{2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}}{b}\right)\right)\right)\right)} \]

7. Simplified 97.6%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{\mathsf{fma}\left(2, c \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{{b}^{2}} \cdot 20\right), 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{{b}^{2}} \cdot 20\right), 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

8. Taylor expanded in c around 0 97.6%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} + c \cdot \left(-42 \cdot \frac{{a}^{3} \cdot c}{{b}^{11}} + -14 \cdot \frac{{a}^{2}}{{b}^{9}}\right)\right)}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

9. Final simplification 97.6%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} + c \cdot \left(-42 \cdot \frac{c \cdot {a}^{3}}{{b}^{11}} + -14 \cdot \frac{{a}^{2}}{{b}^{9}}\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 2: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} + -14 \cdot \frac{c \cdot {a}^{2}}{{b}^{9}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (fma
      -2.0
      (/ (pow c 3.0) (pow b 5.0))
      (*
       (pow c 4.0)
       (+
        (* -5.0 (/ a (pow b 7.0)))
        (* -14.0 (/ (* c (pow a 2.0)) (pow b 9.0)))))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (pow(c, 4.0) * ((-5.0 * (a / pow(b, 7.0))) + (-14.0 * ((c * pow(a, 2.0)) / pow(b, 9.0))))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64((c ^ 4.0) * Float64(Float64(-5.0 * Float64(a / (b ^ 7.0))) + Float64(-14.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 9.0))))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-5.0 * N[(a / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-14.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in a around 0 97.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.25 \cdot \frac{4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + a \cdot \left(-0.25 \cdot \frac{a \cdot \left(2 \cdot \frac{c \cdot \left(2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}} + \left(2 \cdot \frac{{c}^{2} \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{4}} + 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + -0.25 \cdot \frac{2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}}{b}\right)\right)\right)\right)} \]

7. Simplified 97.6%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}}}{b} \cdot 20, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{\mathsf{fma}\left(2, c \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{{b}^{2}} \cdot 20\right), 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}}}{{b}^{2}} \cdot 20\right), 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

8. Taylor expanded in c around 0 97.3%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \color{blue}{{c}^{4} \cdot \left(-14 \cdot \frac{{a}^{2} \cdot c}{{b}^{9}} + -5 \cdot \frac{a}{{b}^{7}}\right)}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

9. Final simplification 97.3%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, {c}^{4} \cdot \left(-5 \cdot \frac{a}{{b}^{7}} + -14 \cdot \frac{c \cdot {a}^{2}}{{b}^{9}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 3: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}\right) - c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (/ (* (pow c 3.0) (pow a 2.0)) (pow b 4.0))
   (-
    (-
     (* -0.25 (* (/ (* (pow c 4.0) (pow a 4.0)) a) (/ 20.0 (pow b 6.0))))
     (* a (pow (/ c (- b)) 2.0)))
    c))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 4.0)), (((-0.25 * (((pow(c, 4.0) * pow(a, 4.0)) / a) * (20.0 / pow(b, 6.0)))) - (a * pow((c / -b), 2.0))) - c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 4.0)), Float64(Float64(Float64(-0.25 * Float64(Float64(Float64((c ^ 4.0) * (a ^ 4.0)) / a) * Float64(20.0 / (b ^ 6.0)))) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) - c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in c around inf 21.1%

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

   1. sqrt-prod 21.3%

      \[\leadsto \frac{\color{blue}{\sqrt{c} \cdot \sqrt{-4 \cdot a + \frac{{b}^{2}}{c}}} - b}{a \cdot 2} \]

   2. fma-neg 22.4%

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

   3. *-commutative 22.4%

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

   4. fma-define 22.4%

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

7. Applied egg-rr 22.4%

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

8. Taylor expanded in b around inf 96.7%

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

10. Simplified 96.7%

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

11. Final simplification 96.7%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}\right) - c\right)}{b} \]
12. Add Preprocessing

Alternative 4: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, -0.25 \cdot \left(a \cdot \left(20 \cdot \frac{\frac{{c}^{4}}{{b}^{6}}}{b}\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (fma
      -2.0
      (/ (pow c 3.0) (pow b 5.0))
      (* -0.25 (* a (* 20.0 (/ (/ (pow c 4.0) (pow b 6.0)) b))))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (-0.25 * (a * (20.0 * ((pow(c, 4.0) / pow(b, 6.0)) / b)))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(-0.25 * Float64(a * Float64(20.0 * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) / b)))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(a * N[(20.0 * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in a around 0 96.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 96.7%

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

   2. mul-1-neg 96.7%

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{\left(-\frac{c}{b}\right)} \]

   3. unsub-neg 96.7%

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) - \frac{c}{b}} \]

7. Simplified 96.7%

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

8. Final simplification 96.7%

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

Alternative 5: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + {c}^{4} \cdot \left(-10 \cdot \frac{{a}^{2}}{{b}^{7}} + -4 \cdot \frac{a}{c \cdot {b}^{5}}\right)\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   a
   (+
    (* -2.0 (/ c b))
    (*
     a
     (+
      (* -2.0 (/ (pow c 2.0) (pow b 3.0)))
      (*
       (pow c 4.0)
       (+
        (* -10.0 (/ (pow a 2.0) (pow b 7.0)))
        (* -4.0 (/ a (* c (pow b 5.0))))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (pow(c, 2.0) / pow(b, 3.0))) + (pow(c, 4.0) * ((-10.0 * (pow(a, 2.0) / pow(b, 7.0))) + (-4.0 * (a / (c * pow(b, 5.0)))))))))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * (((-2.0d0) * (c / b)) + (a * (((-2.0d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((c ** 4.0d0) * (((-10.0d0) * ((a ** 2.0d0) / (b ** 7.0d0))) + ((-4.0d0) * (a / (c * (b ** 5.0d0)))))))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (Math.pow(c, 4.0) * ((-10.0 * (Math.pow(a, 2.0) / Math.pow(b, 7.0))) + (-4.0 * (a / (c * Math.pow(b, 5.0)))))))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (math.pow(c, 4.0) * ((-10.0 * (math.pow(a, 2.0) / math.pow(b, 7.0))) + (-4.0 * (a / (c * math.pow(b, 5.0)))))))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(-2.0 * Float64(c / b)) + Float64(a * Float64(Float64(-2.0 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64((c ^ 4.0) * Float64(Float64(-10.0 * Float64((a ^ 2.0) / (b ^ 7.0))) + Float64(-4.0 * Float64(a / Float64(c * (b ^ 5.0)))))))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * (c / b)) + (a * ((-2.0 * ((c ^ 2.0) / (b ^ 3.0))) + ((c ^ 4.0) * ((-10.0 * ((a ^ 2.0) / (b ^ 7.0))) + (-4.0 * (a / (c * (b ^ 5.0)))))))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-10.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(a / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in a around 0 96.5%

   \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{a \cdot 2} \]

6. Taylor expanded in c around inf 96.5%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + \color{blue}{{c}^{4} \cdot \left(-10 \cdot \frac{{a}^{2}}{{b}^{7}} + -4 \cdot \frac{a}{{b}^{5} \cdot c}\right)}\right)\right)}{a \cdot 2} \]

7. Final simplification 96.5%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + {c}^{4} \cdot \left(-10 \cdot \frac{{a}^{2}}{{b}^{7}} + -4 \cdot \frac{a}{c \cdot {b}^{5}}\right)\right)\right)}{a \cdot 2} \]
8. Add Preprocessing

Alternative 6: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (-
     (*
      c
      (fma
       -2.0
       (/ (pow a 2.0) (pow b 5.0))
       (* -0.25 (* c (* (/ (/ (pow a 4.0) (pow b 6.0)) b) (/ 20.0 a))))))
     (/ a (pow b 3.0))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * ((c * fma(-2.0, (pow(a, 2.0) / pow(b, 5.0)), (-0.25 * (c * (((pow(a, 4.0) / pow(b, 6.0)) / b) * (20.0 / a)))))) - (a / pow(b, 3.0)))) + (-1.0 / b));
}

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(c * fma(-2.0, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(-0.25 * Float64(c * Float64(Float64(Float64((a ^ 4.0) / (b ^ 6.0)) / b) * Float64(20.0 / a)))))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(c * N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(c * N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(20.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in c around inf 21.1%

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

   1. sqrt-prod 21.3%

      \[\leadsto \frac{\color{blue}{\sqrt{c} \cdot \sqrt{-4 \cdot a + \frac{{b}^{2}}{c}}} - b}{a \cdot 2} \]

   2. fma-neg 22.4%

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

   3. *-commutative 22.4%

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

   4. fma-define 22.4%

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

7. Applied egg-rr 22.4%

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

8. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
9. Step-by-step derivation

10. Simplified 96.4%

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

11. Final simplification 96.4%

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

Alternative 7: 96.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((-2.0d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 4.0d0))) - c) - (a * ((c / -b) ** 2.0d0))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (((-2.0 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 4.0))) - c) - (a * Math.pow((c / -b), 2.0))) / b;
}

Python

def code(a, b, c):
	return (((-2.0 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 4.0))) - c) - (a * math.pow((c / -b), 2.0))) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (((-2.0 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 4.0))) - c) - (a * ((c / -b) ^ 2.0))) / b;
end

Wolfram

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

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in c around inf 21.1%

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

   1. sqrt-prod 21.3%

      \[\leadsto \frac{\color{blue}{\sqrt{c} \cdot \sqrt{-4 \cdot a + \frac{{b}^{2}}{c}}} - b}{a \cdot 2} \]

   2. fma-neg 22.4%

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

   3. *-commutative 22.4%

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

   4. fma-define 22.4%

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

7. Applied egg-rr 22.4%

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

8. Taylor expanded in b around inf 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 8: 96.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - (pow(c, 2.0) / pow(b, 3.0)))) - (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 = (a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 5.0))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

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

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in a around 0 95.8%

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

   1. +-commutative 95.8%

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

   2. mul-1-neg 95.8%

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

   3. unsub-neg 95.8%

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

   4. mul-1-neg 95.8%

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

   5. unsub-neg 95.8%

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

7. Simplified 95.8%

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

8. Final simplification 95.8%

   \[\leadsto a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
9. Add Preprocessing

Alternative 9: 96.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
end

Wolfram

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

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in c around 0 95.5%

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

6. Final simplification 95.5%

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

Alternative 10: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in c around inf 21.1%

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

   1. sqrt-prod 21.3%

      \[\leadsto \frac{\color{blue}{\sqrt{c} \cdot \sqrt{-4 \cdot a + \frac{{b}^{2}}{c}}} - b}{a \cdot 2} \]

   2. fma-neg 22.4%

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

   3. *-commutative 22.4%

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

   4. fma-define 22.4%

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

7. Applied egg-rr 22.4%

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

8. Taylor expanded in b around inf 93.8%

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

10. Simplified 93.8%

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

11. Final simplification 93.8%

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

Alternative 11: 90.1% accurate, 29.0× 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 21.1%

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

   1. *-commutative 21.1%

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

3. Simplified 21.1%

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

5. Taylor expanded in b around inf 88.1%

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

   1. associate-*r/ 88.1%

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

   2. mul-1-neg 88.1%

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

7. Simplified 88.1%

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

8. Final simplification 88.1%

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

Reproduce

herbie shell --seed 2024055 
(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)))