Quadratic roots, wide range

Percentage Accurate: 18.1% → 97.5%

Time: 8.0s

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: 18.1% 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

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

FPCore

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

C

double code(double a, double b, double c) {
	return (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 * ((pow(c, 3.0) / pow(b, 5.0)) * (a * a)))) - (c / b)) - (a * (c / (pow(b, 3.0) / c)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

2. Taylor expanded in a around 0 98.0%

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

4. Simplified 98.0%

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

5. Taylor expanded in b around 0 98.0%

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

   1. associate-/l* 98.0%

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

   2. distribute-rgt-out 98.0%

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

   3. metadata-eval 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

   \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right) - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}} \]

Alternative 2: 96.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

2. Taylor expanded in b around inf 97.5%

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

   1. +-commutative 97.5%

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

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. +-commutative 97.5%

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

   5. fma-def 97.5%

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

   6. associate-/l* 97.5%

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

   7. associate-/r/ 97.5%

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

   8. unpow2 97.5%

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

   9. mul-1-neg 97.5%

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

   10. distribute-neg-frac 97.5%

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

   11. associate-/l* 97.5%

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

   12. associate-/r/ 97.5%

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

   13. unpow2 97.5%

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

   14. associate-/l* 97.5%

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

4. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 3: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / ((b ** 3.0d0) / c)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

2. Taylor expanded in b around inf 96.5%

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

   1. +-commutative 96.5%

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

   2. mul-1-neg 96.5%

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

   3. unsub-neg 96.5%

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

   4. mul-1-neg 96.5%

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

   5. distribute-neg-frac 96.5%

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

   6. associate-/l* 96.5%

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

   7. associate-/r/ 96.5%

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

   8. unpow2 96.5%

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

   9. associate-/l* 96.5%

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

4. Simplified 96.5%

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

5. Final simplification 96.5%

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

Alternative 4: 94.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

2. Taylor expanded in b around inf 13.0%

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

   1. flip-+ 13.0%

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

   2. associate-*l/ 13.0%

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

   3. *-commutative 13.0%

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

   4. associate-*l/ 13.0%

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

   5. *-commutative 13.0%

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

   6. associate-*l/ 13.0%

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

   7. *-commutative 13.0%

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

4. Applied egg-rr 13.0%

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

   1. sqr-neg 13.0%

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

6. Simplified 13.0%

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

7. Taylor expanded in b around inf 96.2%

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

   1. *-commutative 96.2%

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

   2. associate-*l* 96.2%

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

9. Simplified 96.2%

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

10. Final simplification 96.2%

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

Alternative 5: 90.2% 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 16.7%

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

2. Taylor expanded in b around inf 91.3%

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

   1. mul-1-neg 91.3%

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

   2. distribute-neg-frac 91.3%

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

4. Simplified 91.3%

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

5. Final simplification 91.3%

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

Reproduce

herbie shell --seed 2023272 
(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)))