Quadratic roots, medium range

Percentage Accurate: 32.0% → 95.4%

Time: 9.8s

Alternatives: 7

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 32.0% 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: 95.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / a) * (20.0d0 / (b ** 7.0d0)))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around inf 96.4%

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

5. Taylor expanded in c around 0 96.4%

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

   1. distribute-rgt-out 96.4%

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

   2. associate-*r* 96.4%

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

   3. *-commutative 96.4%

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

   4. times-frac 96.4%

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

7. Simplified 96.4%

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

8. Final simplification 96.4%

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

Alternative 2: 93.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around inf 94.9%

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

   1. associate-+r+ 94.9%

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

   2. mul-1-neg 94.9%

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

   3. unsub-neg 94.9%

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

   4. mul-1-neg 94.9%

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

   5. unsub-neg 94.9%

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

   6. associate-*r/ 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

6. Simplified 94.9%

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

7. Final simplification 94.9%

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

Alternative 3: 93.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around inf 94.4%

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

   1. fma-def 94.4%

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

   2. cube-prod 94.4%

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

   3. distribute-lft-out 94.4%

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

   4. associate-/l* 94.5%

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

   5. associate-/r/ 94.5%

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

   6. associate-/l* 94.5%

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

6. Simplified 94.5%

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

7. Taylor expanded in a around 0 94.5%

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

   1. unpow2 94.5%

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

   2. unpow2 94.5%

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

   3. swap-sqr 94.5%

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

   4. unpow2 94.5%

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

9. Simplified 94.5%

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

    1. unpow2 94.5%

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

11. Applied egg-rr 94.5%

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

12. Final simplification 94.5%

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

Alternative 4: 90.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around inf 92.0%

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

   1. mul-1-neg 92.0%

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

   2. unsub-neg 92.0%

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

   3. mul-1-neg 92.0%

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

   4. distribute-neg-frac 92.0%

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

   5. associate-/l* 92.0%

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

6. Simplified 92.0%

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

7. Final simplification 92.0%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 / b) - (b / c)) ** (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around inf 91.7%

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

   1. distribute-lft-out 91.7%

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

   2. associate-/l* 91.7%

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

   3. associate-/l* 91.7%

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

6. Simplified 91.7%

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

   1. clear-num 91.6%

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

   2. inv-pow 91.6%

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

   3. +-commutative 91.6%

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

   4. associate-/r/ 91.6%

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

   5. fma-def 91.6%

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

   6. associate-/r/ 91.5%

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

8. Applied egg-rr 91.5%

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

9. Taylor expanded in a around 0 92.0%

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

    1. +-commutative 92.0%

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

    2. mul-1-neg 92.0%

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

    3. unsub-neg 92.0%

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

11. Simplified 92.0%

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

12. Final simplification 92.0%

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

Alternative 6: 80.9% 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 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around inf 82.6%

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

   1. mul-1-neg 82.6%

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

   2. distribute-neg-frac 82.6%

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

6. Simplified 82.6%

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

7. Final simplification 82.6%

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

Alternative 7: 1.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / 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 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

4. Taylor expanded in b around -inf 10.0%

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

   1. +-commutative 10.0%

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

   2. mul-1-neg 10.0%

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

   3. unsub-neg 10.0%

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

6. Simplified 10.0%

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

7. Taylor expanded in c around inf 1.6%

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

8. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023311 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))