Quadratic roots, wide range

Percentage Accurate: 18.2% → 97.5%

Time: 8.6s

Alternatives: 5

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: 18.2% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

   1. neg-sub0 19.0%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 19.0%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 19.0%

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

   4. neg-mul-1 19.0%

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

   5. associate-*l/ 19.0%

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

   6. *-commutative 19.0%

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

   7. associate-/r* 19.0%

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

   8. /-rgt-identity 19.0%

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

   9. metadata-eval 19.0%

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

3. Simplified 19.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in a around 0 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in c around 0 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-/l* 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

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

Alternative 2: 96.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

   1. neg-sub0 19.0%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 19.0%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 19.0%

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

   4. neg-mul-1 19.0%

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

   5. associate-*l/ 19.0%

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

   6. *-commutative 19.0%

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

   7. associate-/r* 19.0%

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

   8. /-rgt-identity 19.0%

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

   9. metadata-eval 19.0%

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

3. Simplified 19.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 96.8%

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

   1. +-commutative 96.8%

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

   2. mul-1-neg 96.8%

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

   3. unsub-neg 96.8%

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

   4. +-commutative 96.8%

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

   5. mul-1-neg 96.8%

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

   6. unsub-neg 96.8%

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

   7. associate-*r/ 96.8%

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

   8. *-commutative 96.8%

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

   9. associate-*r* 96.8%

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

   10. unpow2 96.8%

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

   11. associate-/l* 96.8%

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

   12. unpow2 96.8%

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

6. Simplified 96.8%

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

7. Final simplification 96.8%

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

Alternative 3: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

   1. neg-sub0 19.0%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 19.0%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 19.0%

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

   4. neg-mul-1 19.0%

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

   5. associate-*l/ 19.0%

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

   6. *-commutative 19.0%

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

   7. associate-/r* 19.0%

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

   8. /-rgt-identity 19.0%

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

   9. metadata-eval 19.0%

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

3. Simplified 19.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 94.9%

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

   1. +-commutative 94.9%

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

   2. mul-1-neg 94.9%

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

   3. unsub-neg 94.9%

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

   4. associate-*r/ 94.9%

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

   5. neg-mul-1 94.9%

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

   6. associate-/l* 94.9%

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

   7. unpow2 94.9%

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

6. Simplified 94.9%

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

7. Final simplification 94.9%

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

Alternative 4: 94.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(4 \cdot a\right)}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)} \cdot \frac{-0.5}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ (* c (* 4.0 a)) (+ b (+ b (* -2.0 (* a (/ c b)))))) (/ -0.5 a)))

C

double code(double a, double b, double c) {
	return ((c * (4.0 * a)) / (b + (b + (-2.0 * (a * (c / b)))))) * (-0.5 / 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 = ((c * (4.0d0 * a)) / (b + (b + ((-2.0d0) * (a * (c / b)))))) * ((-0.5d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (4.0 * a)) / (b + (b + (-2.0 * (a * (c / b)))))) * (-0.5 / a);
}

Python

def code(a, b, c):
	return ((c * (4.0 * a)) / (b + (b + (-2.0 * (a * (c / b)))))) * (-0.5 / a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(4.0 * a)) / Float64(b + Float64(b + Float64(-2.0 * Float64(a * Float64(c / b)))))) * Float64(-0.5 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (4.0 * a)) / (b + (b + (-2.0 * (a * (c / b)))))) * (-0.5 / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.0%

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

   1. neg-sub0 19.0%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 19.0%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 19.0%

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

   4. neg-mul-1 19.0%

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

   5. associate-*l/ 19.0%

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

   6. *-commutative 19.0%

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

   7. associate-/r* 19.0%

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

   8. /-rgt-identity 19.0%

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

   9. metadata-eval 19.0%

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

3. Simplified 19.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in a around 0 13.4%

   \[\leadsto \left(b - \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}\right) \cdot \frac{-0.5}{a} \]
5. Step-by-step derivation

   1. *-commutative 13.4%

      \[\leadsto \left(b - \left(b + \color{blue}{\frac{c \cdot a}{b} \cdot -2}\right)\right) \cdot \frac{-0.5}{a} \]

   2. associate-/l* 13.4%

      \[\leadsto \left(b - \left(b + \color{blue}{\frac{c}{\frac{b}{a}}} \cdot -2\right)\right) \cdot \frac{-0.5}{a} \]

6. Simplified 13.4%

   \[\leadsto \left(b - \color{blue}{\left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}\right) \cdot \frac{-0.5}{a} \]
7. Step-by-step derivation

   1. flip-- 13.4%

      \[\leadsto \color{blue}{\frac{b \cdot b - \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right) \cdot \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}} \cdot \frac{-0.5}{a} \]

   2. associate-/l* 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + \color{blue}{\frac{c \cdot a}{b}} \cdot -2\right) \cdot \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   3. *-commutative 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + \color{blue}{-2 \cdot \frac{c \cdot a}{b}}\right) \cdot \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   4. associate-/l* 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right) \cdot \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   5. associate-/r/ 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right) \cdot \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   6. associate-/l* 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + \color{blue}{\frac{c \cdot a}{b}} \cdot -2\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   7. *-commutative 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + \color{blue}{-2 \cdot \frac{c \cdot a}{b}}\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   8. associate-/l* 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   9. associate-/r/ 13.4%

      \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right)}{b + \left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   10. associate-/l* 13.4%

       \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{b + \left(b + \color{blue}{\frac{c \cdot a}{b}} \cdot -2\right)} \cdot \frac{-0.5}{a} \]

   11. *-commutative 13.4%

       \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{b + \left(b + \color{blue}{-2 \cdot \frac{c \cdot a}{b}}\right)} \cdot \frac{-0.5}{a} \]

   12. associate-/l* 13.4%

       \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{b + \left(b + -2 \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right)} \cdot \frac{-0.5}{a} \]

   13. associate-/r/ 13.4%

       \[\leadsto \frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{b + \left(b + -2 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right)} \cdot \frac{-0.5}{a} \]

8. Applied egg-rr 13.4%

   \[\leadsto \color{blue}{\frac{b \cdot b - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right) \cdot \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{b + \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}} \cdot \frac{-0.5}{a} \]

9. Taylor expanded in b around inf 94.6%

   \[\leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{b + \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)} \cdot \frac{-0.5}{a} \]
10. Step-by-step derivation

    1. *-commutative 94.6%

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

    2. associate-*l* 94.6%

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

11. Simplified 94.6%

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

12. Final simplification 94.6%

    \[\leadsto \frac{c \cdot \left(4 \cdot a\right)}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)} \cdot \frac{-0.5}{a} \]

Alternative 5: 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(Float64(-c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

   1. neg-sub0 19.0%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 19.0%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 19.0%

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

   4. neg-mul-1 19.0%

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

   5. associate-*l/ 19.0%

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

   6. *-commutative 19.0%

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

   7. associate-/r* 19.0%

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

   8. /-rgt-identity 19.0%

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

   9. metadata-eval 19.0%

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

3. Simplified 19.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 89.6%

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

   1. associate-*r/ 89.6%

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

   2. neg-mul-1 89.6%

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

6. Simplified 89.6%

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

7. Final simplification 89.6%

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

Reproduce

herbie shell --seed 2023215 
(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)))