Quadratic roots, wide range

Percentage Accurate: 18.3% → 97.5%

Time: 21.0s

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: 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: 97.5% 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 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in b around inf 98.6%

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

7. Simplified 98.6%

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

   1. frac-times 98.6%

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

   2. associate-*r* 98.6%

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

   3. pow-prod-down 98.6%

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

   4. *-commutative 98.6%

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

9. Applied egg-rr 98.6%

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

    1. *-commutative 98.6%

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

    2. associate-*r* 98.6%

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

    3. metadata-eval 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{{c}^{4} \cdot 20}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -0.25 (* a (/ (* (pow c 4.0) 20.0) (pow b 7.0))))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * (a * ((pow(c, 4.0) * 20.0) / pow(b, 7.0)))))) - (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 * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * (a * (((c ** 4.0d0) * 20.0d0) / (b ** 7.0d0)))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-0.25 * (a * ((Math.pow(c, 4.0) * 20.0) / Math.pow(b, 7.0)))))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-0.25 * (a * ((math.pow(c, 4.0) * 20.0) / math.pow(b, 7.0)))))) - (math.pow(c, 2.0) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-0.25 * Float64(a * Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 7.0)))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-0.25 * (a * (((c ^ 4.0) * 20.0) / (b ^ 7.0)))))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 7.0], $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 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in a around 0 98.6%

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

6. Taylor expanded in b around 0 98.6%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\frac{a \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]
7. Step-by-step derivation

   1. associate-/l* 98.6%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(a \cdot \frac{4 \cdot {c}^{4} + 16 \cdot {c}^{4}}{{b}^{7}}\right)}\right)\right) \]

   2. distribute-rgt-out 98.6%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{\color{blue}{{c}^{4} \cdot \left(4 + 16\right)}}{{b}^{7}}\right)\right)\right) \]

   3. metadata-eval 98.6%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{{c}^{4} \cdot \color{blue}{20}}{{b}^{7}}\right)\right)\right) \]

8. Simplified 98.6%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \color{blue}{\left(a \cdot \frac{{c}^{4} \cdot 20}{{b}^{7}}\right)}\right)\right) \]

9. Final simplification 98.6%

   \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \left(a \cdot \frac{{c}^{4} \cdot 20}{{b}^{7}}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 3: 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 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in a around 0 97.8%

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

6. Final simplification 97.8%

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

Alternative 4: 96.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in c around 0 97.4%

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

6. Taylor expanded in b around -inf 97.4%

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

   1. mul-1-neg 97.4%

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

   2. distribute-neg-frac2 97.4%

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

   3. fma-define 97.4%

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

   4. associate-/l* 97.4%

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

   5. unpow2 97.4%

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

   6. unpow2 97.4%

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

   7. unpow2 97.4%

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

   8. times-frac 97.4%

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

   9. swap-sqr 97.4%

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

   10. unpow1 97.4%

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

   11. pow-plus 97.4%

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

   12. associate-*r/ 97.4%

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

   13. metadata-eval 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 5: 95.1% 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(Float64(-c) / 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 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in a around 0 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. mul-1-neg 96.6%

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

   4. distribute-neg-frac2 96.6%

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

   5. associate-/l* 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 6: 95.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in c around 0 96.3%

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

   1. associate-*r/ 96.3%

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

   2. neg-mul-1 96.3%

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

   3. distribute-rgt-neg-in 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in b around inf 96.2%

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

9. Taylor expanded in b around inf 96.6%

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

    1. distribute-lft-out 96.6%

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

    2. associate-*r/ 96.6%

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

    3. mul-1-neg 96.6%

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

    4. distribute-neg-frac2 96.6%

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

11. Simplified 96.6%

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

12. Final simplification 96.6%

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

Alternative 7: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (c * (a / 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) - (c * (a / (b ** 3.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in c around 0 96.3%

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

   1. associate-*r/ 96.3%

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

   2. neg-mul-1 96.3%

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

   3. distribute-rgt-neg-in 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in c around 0 96.3%

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

   1. sub-neg 96.3%

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

   2. associate-*r/ 96.3%

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

   3. mul-1-neg 96.3%

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

   4. distribute-rgt-neg-out 96.3%

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

   5. associate-*r/ 96.3%

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

   6. +-commutative 96.3%

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

   7. associate-*r/ 96.3%

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

   8. distribute-rgt-neg-out 96.3%

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

   9. distribute-frac-neg 96.3%

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

   10. unsub-neg 96.3%

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

   11. distribute-neg-frac 96.3%

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

   12. metadata-eval 96.3%

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

   13. *-commutative 96.3%

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

   14. associate-*r/ 96.3%

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 8: 90.0% 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 13.6%

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

   1. *-commutative 13.6%

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

3. Simplified 13.6%

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

5. Taylor expanded in b around inf 93.2%

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

   1. associate-*r/ 93.2%

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

   2. mul-1-neg 93.2%

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

7. Simplified 93.2%

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

8. Final simplification 93.2%

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

Reproduce

herbie shell --seed 2024096 
(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)))