Quadratic roots, narrow range

Percentage Accurate: 56.2% → 90.7%

Time: 15.1s

Alternatives: 11

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 56.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * ((pow(c, 4.0) * (a * 20.0)) / pow(b, 7.0))))) - (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 = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * (((c ** 4.0d0) * (a * 20.0d0)) / (b ** 7.0d0))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

   1. *-commutative 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in a around 0 93.0%

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

6. Taylor expanded in c around 0 93.0%

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

   1. *-commutative 93.0%

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

   2. associate-*l/ 93.0%

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

   3. associate-*r* 93.0%

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

   4. metadata-eval 93.0%

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

   5. distribute-rgt-out 93.0%

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

   6. *-commutative 93.0%

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

   7. distribute-rgt-out 93.0%

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

   8. metadata-eval 93.0%

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

   9. associate-*l* 93.0%

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

8. Simplified 93.0%

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

9. Final simplification 93.0%

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.75)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (-
     (- (* (* -2.0 (pow a 2.0)) (/ (pow c 3.0) (pow b 4.0))) c)
     (* a (pow (/ c (- b)) 2.0)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.75) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((((-2.0 * pow(a, 2.0)) * (pow(c, 3.0) / pow(b, 4.0))) - c) - (a * pow((c / -b), 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.75)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-2.0 * (a ^ 2.0)) * Float64((c ^ 3.0) / (b ^ 4.0))) - c) - Float64(a * (Float64(c / Float64(-b)) ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.75], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.75

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. +-commutative 84.1%

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

      3. sqr-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. sqr-neg 84.1%

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

      6. fma-neg 84.4%

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

      7. distribute-lft-neg-in 84.4%

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

      8. *-commutative 84.4%

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

      9. *-commutative 84.4%

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

      10. distribute-rgt-neg-in 84.4%

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

      11. metadata-eval 84.4%

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

   3. Simplified 84.4%

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

   if 1.75 < b

   1. Initial program 49.6%

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

      1. *-commutative 49.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in b around inf 93.9%

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

      1. associate-+r+ 93.9%

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

      2. mul-1-neg 93.9%

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

      3. unsub-neg 93.9%

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

      4. mul-1-neg 93.9%

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

      5. unsub-neg 93.9%

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

      6. associate-/l* 93.9%

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

      7. associate-*r* 93.9%

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

   7. Simplified 93.9%

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

      1. associate-/l* 93.9%

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

   9. Applied egg-rr 93.9%

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

   11. Simplified 93.9%

       \[\leadsto \frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \color{blue}{a \cdot {\left(\frac{c}{-b}\right)}^{2}}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.25)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (* -2.0 (/ (* a (pow c 3.0)) (pow b 5.0)))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.25)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.25], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. +-commutative 84.1%

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

      3. sqr-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. sqr-neg 84.1%

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

      6. fma-neg 84.4%

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

      7. distribute-lft-neg-in 84.4%

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

      8. *-commutative 84.4%

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

      9. *-commutative 84.4%

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

      10. distribute-rgt-neg-in 84.4%

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

      11. metadata-eval 84.4%

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

   3. Simplified 84.4%

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

   if 1.25 < b

   1. Initial program 49.6%

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

      1. *-commutative 49.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in a around 0 93.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.18:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - c \cdot \left(\frac{a}{{b}^{3}} - -2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.18)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    c
    (-
     (/ -1.0 b)
     (* c (- (/ a (pow b 3.0)) (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.18) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (c * ((a / pow(b, 3.0)) - (-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.18)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(c * Float64(Float64(a / (b ^ 3.0)) - Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.18], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(c * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.17999999999999994

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. +-commutative 84.1%

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

      3. sqr-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. sqr-neg 84.1%

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

      6. fma-neg 84.4%

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

      7. distribute-lft-neg-in 84.4%

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

      8. *-commutative 84.4%

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

      9. *-commutative 84.4%

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

      10. distribute-rgt-neg-in 84.4%

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

      11. metadata-eval 84.4%

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

   3. Simplified 84.4%

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

   if 1.17999999999999994 < b

   1. Initial program 49.6%

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

      1. *-commutative 49.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in c around 0 93.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.18:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - c \cdot \left(\frac{a}{{b}^{3}} - -2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.55)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (fma a (pow (/ c b) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.55)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.55], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5499999999999998

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. +-commutative 84.1%

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

      3. sqr-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. sqr-neg 84.1%

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

      6. fma-neg 84.3%

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

      7. distribute-lft-neg-in 84.3%

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

      8. *-commutative 84.3%

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

      9. *-commutative 84.3%

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

      10. distribute-rgt-neg-in 84.3%

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

      11. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

   if 2.5499999999999998 < b

   1. Initial program 49.4%

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

      1. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in a around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. unsub-neg 88.4%

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

      3. mul-1-neg 88.4%

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

      4. distribute-neg-frac2 88.4%

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

      5. associate-/l* 88.4%

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

   7. Simplified 88.4%

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

   8. Taylor expanded in a around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. mul-1-neg 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in a around inf 88.2%

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

   13. Simplified 88.3%

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

   14. Taylor expanded in b around inf 88.4%

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

       1. distribute-lft-out 88.4%

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

       2. mul-1-neg 88.4%

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

       3. +-commutative 88.4%

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

       4. associate-/l* 88.4%

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

       5. fma-define 88.4%

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

       6. unpow2 88.4%

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

       7. unpow2 88.4%

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

       8. times-frac 88.4%

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

       9. unpow2 88.4%

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

   16. Simplified 88.4%

       \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.55)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (fma a (pow (/ c b) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.55)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.55], 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], N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5499999999999998

   1. Initial program 84.1%

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

   if 2.5499999999999998 < b

   1. Initial program 49.4%

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

      1. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in a around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. unsub-neg 88.4%

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

      3. mul-1-neg 88.4%

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

      4. distribute-neg-frac2 88.4%

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

      5. associate-/l* 88.4%

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

   7. Simplified 88.4%

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

   8. Taylor expanded in a around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. mul-1-neg 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in a around inf 88.2%

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

   13. Simplified 88.3%

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

   14. Taylor expanded in b around inf 88.4%

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

       1. distribute-lft-out 88.4%

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

       2. mul-1-neg 88.4%

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

       3. +-commutative 88.4%

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

       4. associate-/l* 88.4%

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

       5. fma-define 88.4%

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

       6. unpow2 88.4%

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

       7. unpow2 88.4%

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

       8. times-frac 88.4%

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

       9. unpow2 88.4%

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

   16. Simplified 88.4%

       \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.55)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (* c (- (/ -1.0 b) (/ (* c a) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) / pow(b, 3.0)));
	}
	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 (b <= 2.55d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c * (((-1.0d0) / b) - ((c * a) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.55:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c * ((-1.0 / b) - ((c * a) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.55)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.55)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c * ((-1.0 / b) - ((c * a) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.55], 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], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5499999999999998

   1. Initial program 84.1%

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

   if 2.5499999999999998 < b

   1. Initial program 49.4%

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

      1. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in c around 0 88.3%

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

      1. associate-*r/ 88.3%

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

      2. neg-mul-1 88.3%

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

      3. distribute-rgt-neg-in 88.3%

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

   7. Simplified 88.3%

      \[\leadsto \color{blue}{c \cdot \left(\frac{a \cdot \left(-c\right)}{{b}^{3}} - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (/ (* c a) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) / pow(b, 3.0)));
}

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 * (((-1.0d0) / b) - ((c * a) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((c * a) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((c * a) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

   1. *-commutative 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in c around 0 83.4%

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

   1. associate-*r/ 83.4%

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

   2. neg-mul-1 83.4%

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

   3. distribute-rgt-neg-in 83.4%

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

7. Simplified 83.4%

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

8. Final simplification 83.4%

   \[\leadsto c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right) \]
9. Add Preprocessing

Alternative 9: 80.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

   1. *-commutative 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in a around 0 83.5%

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

   1. mul-1-neg 83.5%

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

   2. unsub-neg 83.5%

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

   3. mul-1-neg 83.5%

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

   4. distribute-neg-frac2 83.5%

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

   5. associate-/l* 83.5%

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

7. Simplified 83.5%

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

8. Taylor expanded in a around inf 83.3%

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

   1. associate-*r/ 83.3%

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

   2. mul-1-neg 83.3%

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

10. Simplified 83.3%

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

11. Taylor expanded in a around inf 83.3%

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

13. Simplified 83.4%

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

    1. unpow2 83.4%

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

15. Applied egg-rr 83.4%

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

16. Final simplification 83.4%

    \[\leadsto a \cdot \frac{\frac{c}{b} \cdot \frac{c}{-b} - \frac{c}{a}}{b} \]
17. Add Preprocessing

Alternative 10: 63.8% 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(c / Float64(-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 54.9%

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

   1. *-commutative 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in b around inf 65.1%

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

   1. associate-*r/ 65.1%

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

   2. mul-1-neg 65.1%

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

7. Simplified 65.1%

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

8. Final simplification 65.1%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 11: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 54.9%

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

   1. *-commutative 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in b around inf 65.1%

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

   1. expm1-log1p-u 59.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right)\right)} \]

   2. expm1-undefine 47.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right)} - 1} \]

   3. associate-/l* 47.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-2 \cdot \frac{\frac{a \cdot c}{b}}{a \cdot 2}}\right)} - 1 \]

   4. associate-/l* 47.7%

      \[\leadsto e^{\mathsf{log1p}\left(-2 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{a \cdot 2}\right)} - 1 \]

7. Applied egg-rr 47.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 47.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)} + \left(-1\right)} \]

   2. metadata-eval 47.7%

      \[\leadsto e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)} + \color{blue}{-1} \]

   3. +-commutative 47.7%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(-2 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 2}\right)}} \]

   4. log1p-undefine 47.7%

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

   5. rem-exp-log 53.3%

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

   6. associate-*r/ 53.3%

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

   7. *-commutative 53.3%

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

   8. times-frac 53.3%

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

   9. metadata-eval 53.3%

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

   10. associate-*r/ 53.3%

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

   11. *-commutative 53.3%

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

   12. associate-/l/ 53.3%

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

   13. associate-*l/ 53.3%

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

   14. neg-mul-1 53.3%

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

   15. unsub-neg 53.3%

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

   16. associate-*l/ 53.3%

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

   17. associate-/l/ 53.3%

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

   18. *-commutative 53.3%

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

   19. associate-*r/ 53.3%

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

9. Simplified 53.3%

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

10. Taylor expanded in c around 0 3.2%

    \[\leadsto -1 + \color{blue}{1} \]

11. Final simplification 3.2%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))