Quadratic roots, wide range

Percentage Accurate: 18.1% → 97.7%

Time: 16.5s

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: 18.1% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

2. Taylor expanded in a around 0 97.2%

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

4. Simplified 97.2%

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

5. Taylor expanded in c around 0 97.2%

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

   1. associate-/l* 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 2: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

2. Taylor expanded in b around inf 96.3%

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

   1. associate-+r+ 96.3%

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

   2. mul-1-neg 96.3%

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

   3. unsub-neg 96.3%

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

   4. mul-1-neg 96.3%

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

   5. unsub-neg 96.3%

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

   6. associate-/l* 96.3%

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

   7. associate-*r/ 96.3%

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

   8. unpow2 96.3%

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

   9. associate-/l* 96.3%

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

   10. associate-/r/ 96.3%

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

   11. unpow2 96.3%

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

4. Simplified 96.3%

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

5. Final simplification 96.3%

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

Alternative 3: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

2. Taylor expanded in b around inf 12.7%

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

   1. flip-+ 12.7%

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

   2. associate-/l* 12.7%

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

   3. associate-/l* 12.7%

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

   4. associate-/l* 12.7%

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

4. Applied egg-rr 12.7%

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

   1. sqr-neg 12.7%

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

   2. associate-/r/ 12.7%

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

   3. associate-/r/ 12.7%

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

   4. associate--r+ 12.7%

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

   5. associate-/r/ 12.7%

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

6. Simplified 12.7%

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

7. Taylor expanded in b around inf 94.3%

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

   1. *-commutative 94.3%

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

   2. associate-*l* 94.3%

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

9. Simplified 94.3%

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

    1. div-inv 94.2%

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

    2. associate-/l* 94.2%

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

    3. cancel-sign-sub-inv 94.2%

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

    4. metadata-eval 94.2%

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

    5. *-commutative 94.2%

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

    6. *-commutative 94.2%

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

    7. *-commutative 94.2%

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

11. Applied egg-rr 94.2%

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

    1. *-commutative 94.2%

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

    2. associate-/r/ 94.2%

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

    3. associate-*l/ 94.2%

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

    4. times-frac 94.3%

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

    5. *-lft-identity 94.3%

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

    6. times-frac 94.8%

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

    7. *-commutative 94.8%

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

13. Simplified 94.8%

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

14. Final simplification 94.8%

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

Alternative 4: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

2. Taylor expanded in b around inf 94.6%

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

   1. mul-1-neg 94.6%

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

   2. unsub-neg 94.6%

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

   3. mul-1-neg 94.6%

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

   4. distribute-neg-frac 94.6%

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

   5. associate-/l* 94.6%

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

   6. associate-/r/ 94.6%

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

   7. unpow2 94.6%

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

4. Simplified 94.6%

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

5. Final simplification 94.6%

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

Alternative 5: 95.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(\left(c \cdot 4\right) \cdot \frac{1}{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* a (* (* c 4.0) (/ 1.0 (- (* 2.0 (* c (/ a b))) (+ b b))))) (* a 2.0)))

C

double code(double a, double b, double c) {
	return (a * ((c * 4.0) * (1.0 / ((2.0 * (c * (a / b))) - (b + 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 = (a * ((c * 4.0d0) * (1.0d0 / ((2.0d0 * (c * (a / b))) - (b + b))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((c * 4.0) * (1.0 / ((2.0 * (c * (a / b))) - (b + b))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (a * ((c * 4.0) * (1.0 / ((2.0 * (c * (a / b))) - (b + b))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(c * 4.0) * Float64(1.0 / Float64(Float64(2.0 * Float64(c * Float64(a / b))) - Float64(b + b))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((c * 4.0) * (1.0 / ((2.0 * (c * (a / b))) - (b + b))))) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 18.5%

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

2. Taylor expanded in b around inf 12.7%

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

   1. flip-+ 12.7%

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

   2. associate-/l* 12.7%

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

   3. associate-/l* 12.7%

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

   4. associate-/l* 12.7%

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

4. Applied egg-rr 12.7%

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

   1. sqr-neg 12.7%

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

   2. associate-/r/ 12.7%

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

   3. associate-/r/ 12.7%

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

   4. associate--r+ 12.7%

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

   5. associate-/r/ 12.7%

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

6. Simplified 12.7%

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

7. Taylor expanded in b around inf 94.3%

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

   1. *-commutative 94.3%

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

   2. associate-*l* 94.3%

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

9. Simplified 94.3%

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

    1. div-inv 94.3%

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

    2. *-commutative 94.3%

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

    3. cancel-sign-sub-inv 94.3%

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

    4. metadata-eval 94.3%

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

    5. *-commutative 94.3%

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

11. Applied egg-rr 94.3%

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

    1. associate-*l* 94.4%

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

13. Simplified 94.4%

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

14. Final simplification 94.4%

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

Alternative 6: 95.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

2. Taylor expanded in b around inf 12.7%

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

   1. flip-+ 12.7%

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

   2. associate-/l* 12.7%

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

   3. associate-/l* 12.7%

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

   4. associate-/l* 12.7%

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

4. Applied egg-rr 12.7%

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

   1. sqr-neg 12.7%

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

   2. associate-/r/ 12.7%

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

   3. associate-/r/ 12.7%

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

   4. associate--r+ 12.7%

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

   5. associate-/r/ 12.7%

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

6. Simplified 12.7%

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

7. Taylor expanded in b around inf 94.3%

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

   1. *-commutative 94.3%

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

   2. associate-*l* 94.3%

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

9. Simplified 94.3%

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

    1. div-inv 94.2%

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

    2. associate-/l* 94.2%

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

    3. cancel-sign-sub-inv 94.2%

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

    4. metadata-eval 94.2%

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

    5. *-commutative 94.2%

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

    6. *-commutative 94.2%

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

    7. *-commutative 94.2%

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

11. Applied egg-rr 94.2%

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

    1. associate-*r/ 94.3%

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

    2. *-rgt-identity 94.3%

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

    3. associate-/r/ 94.3%

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

13. Simplified 94.3%

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

14. Final simplification 94.3%

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

Alternative 7: 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 18.5%

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

2. Taylor expanded in b around inf 89.7%

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

   1. mul-1-neg 89.7%

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

   2. distribute-neg-frac 89.7%

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

4. Simplified 89.7%

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

5. Final simplification 89.7%

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

Alternative 8: 3.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 18.5%

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

   1. add-sqr-sqrt 18.5%

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

   2. difference-of-squares 18.5%

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

   3. associate-*l* 18.5%

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

   4. sqrt-prod 18.5%

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

   5. metadata-eval 18.5%

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

   6. associate-*l* 18.5%

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

   7. sqrt-prod 18.5%

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

   8. metadata-eval 18.5%

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

3. Applied egg-rr 18.5%

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

   1. *-commutative 18.5%

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

   2. cancel-sign-sub-inv 18.5%

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

   3. metadata-eval 18.5%

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

5. Simplified 18.5%

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

6. Taylor expanded in b around inf 3.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.3%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

   2. distribute-rgt-out 3.3%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt 3.3%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

8. Simplified 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Final simplification 3.3%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023283 
(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)))