Quadratic roots, narrow range

Percentage Accurate: 56.1% → 91.1%

Time: 13.3s

Alternatives: 10

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: 56.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + \frac{{a}^{2}}{{b}^{5}} \cdot -2\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.91)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    c
    (+
     (*
      c
      (-
       (*
        c
        (+
         (* -5.0 (/ (* c (pow a 3.0)) (pow b 7.0)))
         (* (/ (pow a 2.0) (pow b 5.0)) -2.0)))
       (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.91) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((c * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 7.0))) + ((pow(a, 2.0) / pow(b, 5.0)) * -2.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.91)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))) + Float64(Float64((a ^ 2.0) / (b ^ 5.0)) * -2.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.91], 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[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.910000000000000031

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. +-commutative 85.5%

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

      3. sqr-neg 85.5%

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

      4. unsub-neg 85.5%

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

      5. sqr-neg 85.5%

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

      6. fma-neg 85.8%

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

      7. distribute-lft-neg-in 85.8%

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

      8. *-commutative 85.8%

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

      9. *-commutative 85.8%

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

      10. distribute-rgt-neg-in 85.8%

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

      11. metadata-eval 85.8%

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

   3. Simplified 85.8%

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

   if -0.910000000000000031 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.0%

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

      1. *-commutative 52.0%

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

      2. +-commutative 52.0%

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

      3. sqr-neg 52.0%

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

      4. unsub-neg 52.0%

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

      5. sqr-neg 52.0%

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

      6. fma-neg 52.1%

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

      7. distribute-lft-neg-in 52.1%

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

      8. *-commutative 52.1%

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

      9. *-commutative 52.1%

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

      10. distribute-rgt-neg-in 52.1%

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

      11. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in a around 0 93.2%

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

   6. Taylor expanded in c around -inf 93.2%

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

      1. expm1-log1p-u 93.2%

         \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

      2. expm1-undefine 67.0%

         \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

   8. Applied egg-rr 67.0%

      \[\leadsto -1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]
   9. Step-by-step derivation

      1. expm1-define 93.2%

         \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

   10. Simplified 93.2%

       \[\leadsto -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

   11. Taylor expanded in c around 0 93.0%

       \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}} + \frac{{a}^{2}}{{b}^{5}} \cdot -2\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. *-commutative 58.1%

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

   2. +-commutative 58.1%

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

   3. sqr-neg 58.1%

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

   4. unsub-neg 58.1%

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

   5. sqr-neg 58.1%

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

   6. fma-neg 58.2%

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

   7. distribute-lft-neg-in 58.2%

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

   8. *-commutative 58.2%

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

   9. *-commutative 58.2%

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

   10. distribute-rgt-neg-in 58.2%

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

   11. metadata-eval 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in a around 0 90.3%

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

6. Taylor expanded in c around -inf 90.3%

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

   1. associate-*r/ 90.3%

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

   2. neg-mul-1 90.3%

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

8. Applied egg-rr 90.3%

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

9. Final simplification 90.3%

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

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\mathsf{fma}\left(a, \frac{1}{b} + \frac{c \cdot a}{{b}^{3}}, \frac{b}{-c}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.91)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (/ 1.0 a)
    (/ (fma a (+ (/ 1.0 b) (/ (* c a) (pow b 3.0))) (/ b (- c))) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.91) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / a) / (fma(a, ((1.0 / b) + ((c * a) / pow(b, 3.0))), (b / -c)) / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.91)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(fma(a, Float64(Float64(1.0 / b) + Float64(Float64(c * a) / (b ^ 3.0))), Float64(b / Float64(-c))) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.91], 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[(1.0 / a), $MachinePrecision] / N[(N[(a * N[(N[(1.0 / b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / (-c)), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.910000000000000031

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. +-commutative 85.5%

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

      3. sqr-neg 85.5%

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

      4. unsub-neg 85.5%

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

      5. sqr-neg 85.5%

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

      6. fma-neg 85.8%

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

      7. distribute-lft-neg-in 85.8%

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

      8. *-commutative 85.8%

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

      9. *-commutative 85.8%

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

      10. distribute-rgt-neg-in 85.8%

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

      11. metadata-eval 85.8%

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

   3. Simplified 85.8%

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

   if -0.910000000000000031 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.0%

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

      1. *-commutative 52.0%

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

      2. +-commutative 52.0%

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

      3. sqr-neg 52.0%

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

      4. unsub-neg 52.0%

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

      5. sqr-neg 52.0%

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

      6. fma-neg 52.1%

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

      7. distribute-lft-neg-in 52.1%

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

      8. *-commutative 52.1%

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

      9. *-commutative 52.1%

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

      10. distribute-rgt-neg-in 52.1%

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

      11. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in c around 0 90.8%

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

      1. clear-num 90.6%

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

      2. inv-pow 90.6%

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

   7. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 90.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      2. associate-/l* 90.6%

         \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      3. associate-/r* 90.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      4. associate-/r* 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      5. associate-*r* 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

      6. *-commutative 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

      7. associate-*l/ 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

   9. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

   10. Taylor expanded in a around 0 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\mathsf{fma}\left(a, \frac{1}{b} + \frac{c \cdot a}{{b}^{3}}, \frac{b}{-c}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b \cdot \left(\left(\frac{\frac{1}{b}}{b} + a \cdot \frac{c}{{b}^{4}}\right) + \frac{-1}{c \cdot a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.91)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (/ 1.0 a)
    (* b (+ (+ (/ (/ 1.0 b) b) (* a (/ c (pow b 4.0)))) (/ -1.0 (* c a)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.91) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / a) / (b * ((((1.0 / b) / b) + (a * (c / pow(b, 4.0)))) + (-1.0 / (c * a))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.91)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(b * Float64(Float64(Float64(Float64(1.0 / b) / b) + Float64(a * Float64(c / (b ^ 4.0)))) + Float64(-1.0 / Float64(c * a)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.91], 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[(1.0 / a), $MachinePrecision] / N[(b * N[(N[(N[(N[(1.0 / b), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.910000000000000031

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. +-commutative 85.5%

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

      3. sqr-neg 85.5%

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

      4. unsub-neg 85.5%

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

      5. sqr-neg 85.5%

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

      6. fma-neg 85.8%

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

      7. distribute-lft-neg-in 85.8%

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

      8. *-commutative 85.8%

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

      9. *-commutative 85.8%

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

      10. distribute-rgt-neg-in 85.8%

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

      11. metadata-eval 85.8%

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

   3. Simplified 85.8%

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

   if -0.910000000000000031 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.0%

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

      1. *-commutative 52.0%

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

      2. +-commutative 52.0%

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

      3. sqr-neg 52.0%

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

      4. unsub-neg 52.0%

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

      5. sqr-neg 52.0%

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

      6. fma-neg 52.1%

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

      7. distribute-lft-neg-in 52.1%

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

      8. *-commutative 52.1%

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

      9. *-commutative 52.1%

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

      10. distribute-rgt-neg-in 52.1%

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

      11. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in c around 0 90.8%

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

      1. clear-num 90.6%

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

      2. inv-pow 90.6%

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

   7. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 90.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      2. associate-/l* 90.6%

         \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      3. associate-/r* 90.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      4. associate-/r* 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      5. associate-*r* 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

      6. *-commutative 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

      7. associate-*l/ 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

   9. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

   10. Taylor expanded in b around inf 91.1%

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

       1. sub-neg 91.1%

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

       2. +-commutative 91.1%

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

       3. mul-1-neg 91.1%

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

       4. distribute-lft1-in 91.1%

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

       5. metadata-eval 91.1%

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

       6. associate-*r/ 91.1%

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

       7. mul-1-neg 91.1%

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

       8. remove-double-neg 91.1%

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

       9. associate-/l* 91.1%

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

       10. distribute-neg-frac 91.1%

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

       11. metadata-eval 91.1%

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

       12. *-commutative 91.1%

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

   12. Simplified 91.1%

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

       1. inv-pow 91.1%

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

       2. unpow2 91.1%

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

       3. unpow-prod-down 91.1%

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

       4. inv-pow 91.1%

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

       5. inv-pow 91.1%

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

   14. Applied egg-rr 91.1%

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

       1. associate-*l/ 91.1%

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

       2. *-lft-identity 91.1%

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

   16. Simplified 91.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b \cdot \left(\left(\frac{\frac{1}{b}}{b} + a \cdot \frac{c}{{b}^{4}}\right) + \frac{-1}{c \cdot a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.91:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b \cdot \left(\left(\frac{\frac{1}{b}}{b} + a \cdot \frac{c}{{b}^{4}}\right) + \frac{-1}{c \cdot a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.91)
     t_0
     (/
      (/ 1.0 a)
      (*
       b
       (+ (+ (/ (/ 1.0 b) b) (* a (/ c (pow b 4.0)))) (/ -1.0 (* c a))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.91) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) / (b * ((((1.0 / b) / b) + (a * (c / pow(b, 4.0)))) + (-1.0 / (c * a))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.91d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / a) / (b * ((((1.0d0 / b) / b) + (a * (c / (b ** 4.0d0)))) + ((-1.0d0) / (c * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.91) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) / (b * ((((1.0 / b) / b) + (a * (c / Math.pow(b, 4.0)))) + (-1.0 / (c * a))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.91:
		tmp = t_0
	else:
		tmp = (1.0 / a) / (b * ((((1.0 / b) / b) + (a * (c / math.pow(b, 4.0)))) + (-1.0 / (c * a))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.91)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / a) / Float64(b * Float64(Float64(Float64(Float64(1.0 / b) / b) + Float64(a * Float64(c / (b ^ 4.0)))) + Float64(-1.0 / Float64(c * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.91)
		tmp = t_0;
	else
		tmp = (1.0 / a) / (b * ((((1.0 / b) / b) + (a * (c / (b ^ 4.0)))) + (-1.0 / (c * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.91], t$95$0, N[(N[(1.0 / a), $MachinePrecision] / N[(b * N[(N[(N[(N[(1.0 / b), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.910000000000000031

   1. Initial program 85.5%

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

   if -0.910000000000000031 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.0%

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

      1. *-commutative 52.0%

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

      2. +-commutative 52.0%

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

      3. sqr-neg 52.0%

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

      4. unsub-neg 52.0%

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

      5. sqr-neg 52.0%

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

      6. fma-neg 52.1%

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

      7. distribute-lft-neg-in 52.1%

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

      8. *-commutative 52.1%

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

      9. *-commutative 52.1%

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

      10. distribute-rgt-neg-in 52.1%

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

      11. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in c around 0 90.8%

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

      1. clear-num 90.6%

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

      2. inv-pow 90.6%

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

   7. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 90.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      2. associate-/l* 90.6%

         \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      3. associate-/r* 90.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      4. associate-/r* 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      5. associate-*r* 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

      6. *-commutative 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

      7. associate-*l/ 90.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

   9. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

   10. Taylor expanded in b around inf 91.1%

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

       1. sub-neg 91.1%

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

       2. +-commutative 91.1%

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

       3. mul-1-neg 91.1%

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

       4. distribute-lft1-in 91.1%

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

       5. metadata-eval 91.1%

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

       6. associate-*r/ 91.1%

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

       7. mul-1-neg 91.1%

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

       8. remove-double-neg 91.1%

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

       9. associate-/l* 91.1%

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

       10. distribute-neg-frac 91.1%

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

       11. metadata-eval 91.1%

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

       12. *-commutative 91.1%

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

   12. Simplified 91.1%

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

       1. inv-pow 91.1%

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

       2. unpow2 91.1%

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

       3. unpow-prod-down 91.1%

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

       4. inv-pow 91.1%

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

       5. inv-pow 91.1%

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

   14. Applied egg-rr 91.1%

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

       1. associate-*l/ 91.1%

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

       2. *-lft-identity 91.1%

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

   16. Simplified 91.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.91:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b \cdot \left(\left(\frac{\frac{1}{b}}{b} + a \cdot \frac{c}{{b}^{4}}\right) + \frac{-1}{c \cdot a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.0035:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b \cdot \left(\frac{1}{{b}^{2}} + \frac{-1}{c \cdot a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.0035)
     t_0
     (/ (/ 1.0 a) (* b (+ (/ 1.0 (pow b 2.0)) (/ -1.0 (* c a))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0035) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) / (b * ((1.0 / pow(b, 2.0)) + (-1.0 / (c * a))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.0035d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / a) / (b * ((1.0d0 / (b ** 2.0d0)) + ((-1.0d0) / (c * a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0035) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) / (b * ((1.0 / Math.pow(b, 2.0)) + (-1.0 / (c * a))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.0035:
		tmp = t_0
	else:
		tmp = (1.0 / a) / (b * ((1.0 / math.pow(b, 2.0)) + (-1.0 / (c * a))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.0035)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / a) / Float64(b * Float64(Float64(1.0 / (b ^ 2.0)) + Float64(-1.0 / Float64(c * a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.0035)
		tmp = t_0;
	else
		tmp = (1.0 / a) / (b * ((1.0 / (b ^ 2.0)) + (-1.0 / (c * a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0035], t$95$0, N[(N[(1.0 / a), $MachinePrecision] / N[(b * N[(N[(1.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00350000000000000007

   1. Initial program 79.5%

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

   if -0.00350000000000000007 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

      2. +-commutative 44.8%

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

      3. sqr-neg 44.8%

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

      4. unsub-neg 44.8%

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

      5. sqr-neg 44.8%

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

      6. fma-neg 44.8%

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

      7. distribute-lft-neg-in 44.8%

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

      8. *-commutative 44.8%

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

      9. *-commutative 44.8%

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

      10. distribute-rgt-neg-in 44.8%

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

      11. metadata-eval 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in c around 0 94.8%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

   7. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 94.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      2. associate-/l* 94.6%

         \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      3. associate-/r* 94.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      4. associate-/r* 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      5. associate-*r* 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

      6. *-commutative 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

      7. associate-*l/ 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

   9. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

   10. Taylor expanded in b around inf 91.4%

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

       1. *-commutative 91.4%

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

   12. Simplified 91.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{b \cdot \left(\frac{1}{{b}^{2}} + \frac{-1}{c \cdot a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.0035:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.0035) t_0 (/ (/ 1.0 a) (/ (- (/ a b) (/ b c)) a)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0035) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) / (((a / b) - (b / c)) / a);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.0035d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / a) / (((a / b) - (b / c)) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0035) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) / (((a / b) - (b / c)) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.0035:
		tmp = t_0
	else:
		tmp = (1.0 / a) / (((a / b) - (b / c)) / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.0035)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(a / b) - Float64(b / c)) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.0035)
		tmp = t_0;
	else
		tmp = (1.0 / a) / (((a / b) - (b / c)) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0035], t$95$0, N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00350000000000000007

   1. Initial program 79.5%

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

   if -0.00350000000000000007 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

      2. +-commutative 44.8%

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

      3. sqr-neg 44.8%

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

      4. unsub-neg 44.8%

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

      5. sqr-neg 44.8%

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

      6. fma-neg 44.8%

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

      7. distribute-lft-neg-in 44.8%

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

      8. *-commutative 44.8%

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

      9. *-commutative 44.8%

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

      10. distribute-rgt-neg-in 44.8%

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

      11. metadata-eval 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in c around 0 94.8%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

   7. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 94.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      2. associate-/l* 94.6%

         \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      3. associate-/r* 94.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      4. associate-/r* 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

      5. associate-*r* 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

      6. *-commutative 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

      7. associate-*l/ 94.6%

         \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

   9. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

   10. Taylor expanded in a around 0 91.4%

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

       1. +-commutative 91.4%

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

       2. mul-1-neg 91.4%

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

       3. unsub-neg 91.4%

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

   12. Simplified 91.4%

       \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.0035:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{a}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (1.0d0 / a) / (((c / b) - (b / a)) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. *-commutative 58.1%

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

   2. +-commutative 58.1%

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

   3. sqr-neg 58.1%

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

   4. unsub-neg 58.1%

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

   5. sqr-neg 58.1%

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

   6. fma-neg 58.2%

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

   7. distribute-lft-neg-in 58.2%

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

   8. *-commutative 58.2%

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

   9. *-commutative 58.2%

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

   10. distribute-rgt-neg-in 58.2%

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

   11. metadata-eval 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in c around 0 86.6%

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

   1. clear-num 86.5%

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

   2. inv-pow 86.5%

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

7. Applied egg-rr 86.5%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 86.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   2. associate-/l* 86.4%

      \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   3. associate-/r* 86.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   4. associate-/r* 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   5. associate-*r* 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

   6. *-commutative 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

   7. associate-*l/ 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

9. Simplified 86.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

10. Taylor expanded in c around 0 80.6%

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

    1. +-commutative 80.6%

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

    2. mul-1-neg 80.6%

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

    3. unsub-neg 80.6%

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

12. Simplified 80.6%

    \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}} \]
13. Add Preprocessing

Alternative 9: 81.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{a}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. *-commutative 58.1%

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

   2. +-commutative 58.1%

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

   3. sqr-neg 58.1%

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

   4. unsub-neg 58.1%

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

   5. sqr-neg 58.1%

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

   6. fma-neg 58.2%

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

   7. distribute-lft-neg-in 58.2%

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

   8. *-commutative 58.2%

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

   9. *-commutative 58.2%

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

   10. distribute-rgt-neg-in 58.2%

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

   11. metadata-eval 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in c around 0 86.6%

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

   1. clear-num 86.5%

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

   2. inv-pow 86.5%

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

7. Applied egg-rr 86.5%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 86.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   2. associate-/l* 86.4%

      \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   3. associate-/r* 86.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{2}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   4. associate-/r* 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, -2 \cdot \left({a}^{2} \cdot {b}^{-3}\right)\right), \frac{a}{b} \cdot -2\right)}}} \]

   5. associate-*r* 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot {b}^{-3}}\right), \frac{a}{b} \cdot -2\right)}} \]

   6. *-commutative 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \color{blue}{\left({a}^{2} \cdot -2\right)} \cdot {b}^{-3}\right), \frac{a}{b} \cdot -2\right)}} \]

   7. associate-*l/ 86.5%

      \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \color{blue}{\frac{a \cdot -2}{b}}\right)}} \]

9. Simplified 86.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{\frac{2}{c}}{\mathsf{fma}\left(c, \mathsf{fma}\left(-4, {a}^{3} \cdot \frac{c}{{b}^{5}}, \left({a}^{2} \cdot -2\right) \cdot {b}^{-3}\right), \frac{a \cdot -2}{b}\right)}}} \]

10. Taylor expanded in a around 0 80.5%

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

    1. +-commutative 80.5%

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

    2. mul-1-neg 80.5%

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

    3. unsub-neg 80.5%

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

12. Simplified 80.5%

    \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}} \]
13. 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(-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 58.1%

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

   1. *-commutative 58.1%

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

   2. +-commutative 58.1%

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

   3. sqr-neg 58.1%

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

   4. unsub-neg 58.1%

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

   5. sqr-neg 58.1%

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

   6. fma-neg 58.2%

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

   7. distribute-lft-neg-in 58.2%

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

   8. *-commutative 58.2%

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

   9. *-commutative 58.2%

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

   10. distribute-rgt-neg-in 58.2%

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

   11. metadata-eval 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in b around inf 62.4%

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

   1. associate-*r/ 62.4%

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

   2. mul-1-neg 62.4%

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

7. Simplified 62.4%

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

8. Final simplification 62.4%

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

Reproduce

herbie shell --seed 2024129 
(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)))