Quadratic roots, medium range

Percentage Accurate: 31.6% → 95.2%

Time: 13.2s

Alternatives: 8

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: 31.6% 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: 95.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in b around inf 94.1%

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

   1. +-commutative 94.1%

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

   2. mul-1-neg 94.1%

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

   3. unsub-neg 94.1%

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

6. Simplified 94.1%

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

7. Taylor expanded in c around 0 94.1%

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

   1. distribute-rgt-out 94.1%

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

   2. associate-*r* 94.1%

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

   3. *-commutative 94.1%

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

   4. times-frac 94.1%

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

9. Simplified 94.1%

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

10. Final simplification 94.1%

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

Alternative 2: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in b around inf 92.4%

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

   1. +-commutative 92.4%

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

   2. mul-1-neg 92.4%

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

   3. unsub-neg 92.4%

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

   4. +-commutative 92.4%

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

   5. mul-1-neg 92.4%

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

   6. unsub-neg 92.4%

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

   7. associate-*r/ 92.4%

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

   8. *-commutative 92.4%

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

   9. associate-*r* 92.4%

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

   10. unpow2 92.4%

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

   11. associate-/l* 92.4%

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

   12. unpow2 92.4%

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

6. Simplified 92.4%

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

7. Final simplification 92.4%

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

Alternative 3: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.3:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-c\right)}{\frac{{b}^{3}}{a}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.3)
   (* (- b (sqrt (+ (* b b) (* a (* c -4.0))))) (/ -0.5 a))
   (- (/ (* c (- c)) (/ (pow b 3.0) a)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.3) {
		tmp = (b - sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = ((c * -c) / (pow(b, 3.0) / a)) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-0.3d0)) then
        tmp = (b - sqrt(((b * b) + (a * (c * (-4.0d0)))))) * ((-0.5d0) / a)
    else
        tmp = ((c * -c) / ((b ** 3.0d0) / a)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.3) {
		tmp = (b - Math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = ((c * -c) / (Math.pow(b, 3.0) / a)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.3:
		tmp = (b - math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a)
	else:
		tmp = ((c * -c) / (math.pow(b, 3.0) / a)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.3)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(c * Float64(-c)) / Float64((b ^ 3.0) / a)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.3)
		tmp = (b - sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	else
		tmp = ((c * -c) / ((b ^ 3.0) / a)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.3], N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * (-c)), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.299999999999999989

   1. Initial program 76.3%

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

      1. neg-sub0 76.3%

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

      2. associate-+l- 76.3%

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

      3. sub0-neg 76.3%

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

      4. neg-mul-1 76.3%

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

      5. associate-*l/ 76.3%

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

      6. *-commutative 76.3%

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

      7. associate-/r* 76.3%

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

      8. /-rgt-identity 76.3%

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

      9. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

      1. fma-udef 76.3%

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

   5. Applied egg-rr 76.3%

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

   if -0.299999999999999989 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 23.9%

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

      1. neg-sub0 23.9%

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

      2. associate-+l- 23.9%

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

      3. sub0-neg 23.9%

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

      4. neg-mul-1 23.9%

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

      5. associate-*l/ 23.9%

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

      6. *-commutative 23.9%

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

      7. associate-/r* 23.9%

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

      8. /-rgt-identity 23.9%

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

      9. metadata-eval 23.9%

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

   3. Simplified 23.9%

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

   4. Taylor expanded in b around inf 94.0%

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

      1. +-commutative 94.0%

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

      2. mul-1-neg 94.0%

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

      3. unsub-neg 94.0%

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

      4. associate-*r/ 94.0%

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

      5. neg-mul-1 94.0%

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

      6. associate-/l* 94.0%

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

      7. unpow2 94.0%

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

   6. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.3:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-c\right)}{\frac{{b}^{3}}{a}} - \frac{c}{b}\\ \end{array} \]

Alternative 4: 90.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / a), $MachinePrecision] * 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]), $MachinePrecision]

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in a around 0 20.9%

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

   1. *-commutative 20.9%

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

   2. associate-/l* 20.9%

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

6. Simplified 20.9%

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

   1. flip-- 20.7%

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

   2. associate-/l* 20.7%

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

   3. *-commutative 20.7%

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

   4. associate-/l* 20.7%

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

   5. associate-/r/ 20.7%

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

   6. associate-/l* 20.7%

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

   7. *-commutative 20.7%

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

   8. associate-/l* 20.7%

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

   9. associate-/r/ 20.7%

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

   10. associate-/l* 20.7%

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

   11. *-commutative 20.7%

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

   12. associate-/l* 20.7%

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

   13. associate-/r/ 20.7%

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

8. Applied egg-rr 20.7%

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

9. Taylor expanded in b around inf 89.2%

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

    1. *-commutative 89.2%

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

    2. associate-*r* 89.2%

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

    3. *-commutative 89.2%

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

11. Simplified 89.2%

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

12. Final simplification 89.2%

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

Alternative 5: 90.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in a around 0 20.9%

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

   1. *-commutative 20.9%

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

   2. associate-/l* 20.9%

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

6. Simplified 20.9%

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

   1. flip-- 20.7%

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

   2. associate-/l* 20.7%

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

   3. *-commutative 20.7%

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

   4. associate-/l* 20.7%

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

   5. associate-/r/ 20.7%

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

   6. associate-/l* 20.7%

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

   7. *-commutative 20.7%

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

   8. associate-/l* 20.7%

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

   9. associate-/r/ 20.7%

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

   10. associate-/l* 20.7%

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

   11. *-commutative 20.7%

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

   12. associate-/l* 20.7%

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

   13. associate-/r/ 20.7%

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

8. Applied egg-rr 20.7%

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

9. Taylor expanded in b around inf 89.2%

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

    1. *-commutative 89.2%

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

    2. associate-*r* 89.2%

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

    3. *-commutative 89.2%

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

11. Simplified 89.2%

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

12. Taylor expanded in b around 0 89.2%

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

13. Final simplification 89.2%

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

Alternative 6: 90.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in a around 0 20.9%

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

   1. *-commutative 20.9%

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

   2. associate-/l* 20.9%

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

6. Simplified 20.9%

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

   1. flip-- 20.7%

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

   2. associate-/l* 20.7%

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

   3. *-commutative 20.7%

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

   4. associate-/l* 20.7%

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

   5. associate-/r/ 20.7%

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

   6. associate-/l* 20.7%

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

   7. *-commutative 20.7%

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

   8. associate-/l* 20.7%

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

   9. associate-/r/ 20.7%

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

   10. associate-/l* 20.7%

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

   11. *-commutative 20.7%

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

   12. associate-/l* 20.7%

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

   13. associate-/r/ 20.7%

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

8. Applied egg-rr 20.7%

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

9. Taylor expanded in b around inf 89.2%

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

    1. *-commutative 89.2%

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

    2. associate-*r* 89.2%

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

    3. *-commutative 89.2%

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

11. Simplified 89.2%

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

    1. associate-*r/ 89.3%

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

    2. associate-/l* 89.3%

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

    3. associate-/r/ 89.3%

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

    4. div-inv 89.3%

       \[\leadsto \frac{\frac{c}{\frac{b + \left(b + -2 \cdot \color{blue}{\left(c \cdot \frac{1}{\frac{b}{a}}\right)}\right)}{4 \cdot a}} \cdot -0.5}{a} \]

    5. clear-num 89.3%

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

    6. *-commutative 89.3%

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

13. Applied egg-rr 89.3%

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

14. Final simplification 89.3%

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

Alternative 7: 81.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in b around inf 80.3%

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

   1. associate-*r/ 80.3%

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

   2. neg-mul-1 80.3%

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

6. Simplified 80.3%

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

7. Final simplification 80.3%

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

Alternative 8: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

   1. neg-sub0 32.3%

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

   2. associate-+l- 32.3%

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

   3. sub0-neg 32.3%

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

   4. neg-mul-1 32.3%

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

   5. associate-*l/ 32.3%

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

   6. *-commutative 32.3%

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

   7. associate-/r* 32.3%

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

   8. /-rgt-identity 32.3%

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

   9. metadata-eval 32.3%

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

3. Simplified 32.3%

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

4. Taylor expanded in a around 0 20.9%

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

   1. *-commutative 20.9%

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

   2. associate-/l* 20.9%

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

6. Simplified 20.9%

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

   1. flip-- 20.7%

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

   2. associate-/l* 20.7%

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

   3. *-commutative 20.7%

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

   4. associate-/l* 20.7%

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

   5. associate-/r/ 20.7%

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

   6. associate-/l* 20.7%

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

   7. *-commutative 20.7%

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

   8. associate-/l* 20.7%

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

   9. associate-/r/ 20.7%

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

   10. associate-/l* 20.7%

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

   11. *-commutative 20.7%

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

   12. associate-/l* 20.7%

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

   13. associate-/r/ 20.7%

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

8. Applied egg-rr 20.7%

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

9. Taylor expanded in b around inf 89.2%

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

    1. *-commutative 89.2%

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

    2. associate-*r* 89.2%

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

    3. *-commutative 89.2%

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

11. Simplified 89.2%

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

12. Taylor expanded in c around inf 1.6%

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

13. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023178 
(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)))