Quadratic roots, wide range

Percentage Accurate: 17.9% → 97.6%

Time: 19.6s

Alternatives: 8

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: 17.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in b around inf 96.6%

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

7. Simplified 96.6%

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

   1. frac-times 96.6%

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

   2. associate-*r* 96.6%

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

   3. pow-prod-down 96.6%

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

   4. *-commutative 96.6%

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

9. Applied egg-rr 96.6%

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

    1. *-commutative 96.6%

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

    2. associate-*r* 96.6%

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

    3. metadata-eval 96.6%

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

11. Simplified 96.6%

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

12. Final simplification 96.6%

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

Alternative 2: 97.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * fma(-1.0, Float64(a / (b ^ 3.0)), Float64(c * fma(-2.0, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(-0.25 * Float64(c * Float64(Float64(20.0 * (a ^ 3.0)) / (b ^ 7.0)))))))) + Float64(-1.0 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + 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[(20.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in c around 0 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in a around 0 96.3%

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

   1. associate-*r/ 96.3%

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 3: 96.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((((pow(c, 3.0) / pow(b, 4.0)) * (-2.0 * pow(a, 2.0))) - 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 = (((((c ** 3.0d0) / (b ** 4.0d0)) * ((-2.0d0) * (a ** 2.0d0))) - c) - (a * ((c / -b) ** 2.0d0))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in b around inf 95.6%

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

   1. associate-+r+ 95.6%

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

   2. mul-1-neg 95.6%

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

   3. unsub-neg 95.6%

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

   4. mul-1-neg 95.6%

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

   5. unsub-neg 95.6%

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

   6. associate-/l* 95.6%

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

   7. associate-*r* 95.6%

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

7. Simplified 95.6%

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

   1. log1p-expm1-u 66.6%

      \[\leadsto \frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(a \cdot {c}^{2}\right)\right)}}{{b}^{2}}}{b} \]

   2. log1p-undefine 64.1%

      \[\leadsto \frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(a \cdot {c}^{2}\right)\right)}}{{b}^{2}}}{b} \]

9. Applied egg-rr 64.1%

   \[\leadsto \frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(a \cdot {c}^{2}\right)\right)}}{{b}^{2}}}{b} \]

10. Taylor expanded in a around 0 95.6%

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

    1. associate-/l* 95.6%

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

    2. unpow2 95.6%

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

    3. unpow2 95.6%

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

    4. times-frac 95.6%

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

    5. sqr-neg 95.6%

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

    6. mul-1-neg 95.6%

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

    7. associate-*r/ 95.6%

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

    8. *-commutative 95.6%

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

    9. mul-1-neg 95.6%

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

    10. associate-*r/ 95.6%

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

    11. *-commutative 95.6%

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

    12. unpow2 95.6%

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

    13. *-commutative 95.6%

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

    14. mul-1-neg 95.6%

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

12. Simplified 95.6%

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

13. Final simplification 95.6%

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

Alternative 4: 96.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* (pow a 2.0) c) (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 * ((pow(a, 2.0) * c) / 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) * (((a ** 2.0d0) * c) / (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 * ((Math.pow(a, 2.0) * c) / 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 * ((math.pow(a, 2.0) * c) / 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((a ^ 2.0) * c) / (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 * (((a ^ 2.0) * c) / (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[(N[Power[a, 2.0], $MachinePrecision] * c), $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 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in c around 0 95.3%

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

6. Final simplification 95.3%

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

Alternative 5: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in a around 0 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. mul-1-neg 93.7%

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

   4. distribute-neg-frac2 93.7%

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

   5. associate-/l* 93.7%

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

7. Simplified 93.7%

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

8. Final simplification 93.7%

   \[\leadsto \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \]
9. Add Preprocessing

Alternative 6: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{-b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in b around inf 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. mul-1-neg 93.7%

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

7. Simplified 93.7%

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

8. Final simplification 93.7%

   \[\leadsto \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{-b} \]
9. Add Preprocessing

Alternative 7: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in c around 0 93.4%

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

   1. associate-*r/ 93.4%

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

   2. neg-mul-1 93.4%

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

   3. distribute-rgt-neg-in 93.4%

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

7. Simplified 93.4%

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

8. Final simplification 93.4%

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

Alternative 8: 90.3% 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 19.8%

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

   1. *-commutative 19.8%

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

3. Simplified 19.8%

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

5. Taylor expanded in b around inf 88.5%

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

   1. associate-*r/ 88.5%

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

   2. mul-1-neg 88.5%

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

7. Simplified 88.5%

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

8. Final simplification 88.5%

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

Reproduce

herbie shell --seed 2024067 
(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)))