Quadratic roots, narrow range

Percentage Accurate: 56.1% → 99.3%

Time: 15.0s

Alternatives: 8

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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (* (/ 0.5 c) (- (- b) (sqrt (fma c (* a -4.0) (pow b 2.0)))))))

C

double code(double a, double b, double c) {
	return 1.0 / ((0.5 / c) * (-b - sqrt(fma(c, (a * -4.0), pow(b, 2.0)))));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(0.5 / c) * Float64(Float64(-b) - sqrt(fma(c, Float64(a * -4.0), (b ^ 2.0))))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(0.5 / c), $MachinePrecision] * N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. *-commutative 54.0%

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

3. Simplified 54.0%

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

6. Applied egg-rr 53.7%

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

   1. flip-+ 53.4%

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

8. Applied egg-rr 55.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. unpow2 55.0%

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

   2. sqr-neg 55.0%

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

   3. unpow2 55.0%

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

   4. remove-double-div 55.3%

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

   5. fma-udef 55.7%

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

   6. unpow2 55.7%

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

   7. *-commutative 55.7%

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

   8. associate-*r* 55.7%

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

   9. +-commutative 55.7%

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

   10. associate-*r* 55.7%

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

   11. *-commutative 55.7%

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

   12. fma-def 55.7%

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

   13. *-commutative 55.7%

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

10. Simplified 55.7%

    \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. clear-num 55.7%

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

    2. inv-pow 55.7%

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

    3. inv-pow 55.7%

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

    4. pow-pow 55.7%

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

    5. metadata-eval 55.7%

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

    6. pow1/2 55.7%

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

12. Applied egg-rr 55.7%

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

    1. unpow-1 55.7%

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

    2. associate-/r/ 55.7%

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

14. Simplified 55.7%

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

15. Taylor expanded in a around 0 99.4%

    \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)} \]

16. Final simplification 99.4%

    \[\leadsto \frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)} \]
17. Add Preprocessing

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -120:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} - \frac{b}{c}\right) + \frac{{a}^{2}}{\frac{{b}^{3}}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -120.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (+ (- (/ a b) (/ b c)) (/ (pow a 2.0) (/ (pow b 3.0) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -120.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (((a / b) - (b / c)) + (pow(a, 2.0) / (pow(b, 3.0) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -120.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(a / b) - Float64(b / c)) + Float64((a ^ 2.0) / Float64((b ^ 3.0) / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -120.0], 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[(1.0 / N[(N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / c), $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 4 a) c)))) (*.f64 2 a)) < -120

   1. Initial program 93.0%

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

      1. sqr-neg 93.0%

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

      2. +-commutative 93.0%

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

      3. unsub-neg 93.0%

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

      4. sqr-neg 93.0%

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

      5. fma-neg 93.0%

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

      6. distribute-lft-neg-in 93.0%

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

      7. *-commutative 93.0%

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

      8. *-commutative 93.0%

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

      9. distribute-rgt-neg-in 93.0%

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

      10. metadata-eval 93.0%

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

      11. *-commutative 93.0%

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

   3. Simplified 93.0%

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

   if -120 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   6. Applied egg-rr 51.5%

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

      1. flip-+ 51.2%

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

   8. Applied egg-rr 52.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. unpow2 52.8%

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

      2. sqr-neg 52.8%

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

      3. unpow2 52.8%

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

      4. remove-double-div 53.1%

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

      5. fma-udef 53.5%

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

      6. unpow2 53.5%

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

      7. *-commutative 53.5%

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

      8. associate-*r* 53.5%

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

      9. +-commutative 53.5%

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

      10. associate-*r* 53.5%

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

      11. *-commutative 53.5%

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

      12. fma-def 53.5%

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

      13. *-commutative 53.5%

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

   10. Simplified 53.5%

       \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 53.5%

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

       2. inv-pow 53.5%

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

       3. inv-pow 53.5%

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

       4. pow-pow 53.5%

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

       5. metadata-eval 53.5%

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

       6. pow1/2 53.5%

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

   12. Applied egg-rr 53.5%

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

       1. unpow-1 53.5%

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

       2. associate-/r/ 53.5%

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

   14. Simplified 53.5%

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

   15. Taylor expanded in a around 0 91.2%

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

       1. associate-+r+ 91.1%

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

       2. +-commutative 91.1%

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

       3. mul-1-neg 91.1%

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

       4. unsub-neg 91.1%

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

       5. associate-/l* 91.1%

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

   17. Simplified 91.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -120:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} - \frac{b}{c}\right) + \frac{{a}^{2}}{\frac{{b}^{3}}{c}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -120:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} + \frac{c \cdot {a}^{2}}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -120.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (- (+ (/ a b) (/ (* c (pow a 2.0)) (pow b 3.0))) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -120.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (((a / b) + ((c * pow(a, 2.0)) / pow(b, 3.0))) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -120.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(a / b) + Float64(Float64(c * (a ^ 2.0)) / (b ^ 3.0))) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -120.0], 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[(1.0 / N[(N[(N[(a / b), $MachinePrecision] + N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $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 4 a) c)))) (*.f64 2 a)) < -120

   1. Initial program 93.0%

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

      1. sqr-neg 93.0%

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

      2. +-commutative 93.0%

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

      3. unsub-neg 93.0%

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

      4. sqr-neg 93.0%

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

      5. fma-neg 93.0%

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

      6. distribute-lft-neg-in 93.0%

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

      7. *-commutative 93.0%

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

      8. *-commutative 93.0%

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

      9. distribute-rgt-neg-in 93.0%

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

      10. metadata-eval 93.0%

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

      11. *-commutative 93.0%

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

   3. Simplified 93.0%

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

   if -120 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   6. Applied egg-rr 51.5%

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

      1. flip-+ 51.2%

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

   8. Applied egg-rr 52.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. unpow2 52.8%

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

      2. sqr-neg 52.8%

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

      3. unpow2 52.8%

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

      4. remove-double-div 53.1%

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

      5. fma-udef 53.5%

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

      6. unpow2 53.5%

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

      7. *-commutative 53.5%

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

      8. associate-*r* 53.5%

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

      9. +-commutative 53.5%

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

      10. associate-*r* 53.5%

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

      11. *-commutative 53.5%

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

      12. fma-def 53.5%

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

      13. *-commutative 53.5%

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

   10. Simplified 53.5%

       \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 53.5%

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

       2. inv-pow 53.5%

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

       3. inv-pow 53.5%

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

       4. pow-pow 53.5%

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

       5. metadata-eval 53.5%

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

       6. pow1/2 53.5%

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

   12. Applied egg-rr 53.5%

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

       1. unpow-1 53.5%

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

       2. associate-/r/ 53.5%

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

   14. Simplified 53.5%

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

   15. Taylor expanded in a around 0 91.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -120:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} + \frac{c \cdot {a}^{2}}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.08)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.08) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.08)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.08], 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[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $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 4 a) c)))) (*.f64 2 a)) < -0.0800000000000000017

   1. Initial program 79.0%

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

      1. sqr-neg 79.0%

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

      2. +-commutative 79.0%

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

      3. unsub-neg 79.0%

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

      4. sqr-neg 79.0%

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

      5. fma-neg 79.1%

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

      6. distribute-lft-neg-in 79.1%

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

      7. *-commutative 79.1%

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

      8. *-commutative 79.1%

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

      9. distribute-rgt-neg-in 79.1%

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

      10. metadata-eval 79.1%

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

      11. *-commutative 79.1%

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

   3. Simplified 79.1%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

   3. Simplified 46.8%

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

   6. Applied egg-rr 46.5%

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

      1. flip-+ 46.2%

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

   8. Applied egg-rr 47.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. unpow2 47.7%

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

      2. sqr-neg 47.7%

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

      3. unpow2 47.7%

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

      4. remove-double-div 48.1%

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

      5. fma-udef 48.5%

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

      6. unpow2 48.5%

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

      7. *-commutative 48.5%

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

      8. associate-*r* 48.5%

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

      9. +-commutative 48.5%

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

      10. associate-*r* 48.5%

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

      11. *-commutative 48.5%

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

      12. fma-def 48.5%

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

      13. *-commutative 48.5%

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

   10. Simplified 48.5%

       \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 48.5%

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

       2. inv-pow 48.5%

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

       3. inv-pow 48.5%

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

       4. pow-pow 48.5%

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

       5. metadata-eval 48.5%

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

       6. pow1/2 48.5%

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

   12. Applied egg-rr 48.5%

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

       1. unpow-1 48.5%

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

       2. associate-/r/ 48.5%

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

   14. Simplified 48.5%

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

   15. Taylor expanded in a around 0 88.8%

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

       1. +-commutative 88.8%

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

       2. mul-1-neg 88.8%

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

       3. unsub-neg 88.8%

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

   17. Simplified 88.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.08:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.08) t_0 (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.08) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.08d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    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 * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.08) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.08:
		tmp = t_0
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.08)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.08)
		tmp = t_0;
	else
		tmp = 1.0 / ((a / b) - (b / c));
	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[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.08], t$95$0, N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $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 4 a) c)))) (*.f64 2 a)) < -0.0800000000000000017

   1. Initial program 79.0%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

   3. Simplified 46.8%

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

   6. Applied egg-rr 46.5%

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

      1. flip-+ 46.2%

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

   8. Applied egg-rr 47.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. unpow2 47.7%

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

      2. sqr-neg 47.7%

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

      3. unpow2 47.7%

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

      4. remove-double-div 48.1%

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

      5. fma-udef 48.5%

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

      6. unpow2 48.5%

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

      7. *-commutative 48.5%

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

      8. associate-*r* 48.5%

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

      9. +-commutative 48.5%

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

      10. associate-*r* 48.5%

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

      11. *-commutative 48.5%

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

      12. fma-def 48.5%

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

      13. *-commutative 48.5%

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

   10. Simplified 48.5%

       \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 48.5%

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

       2. inv-pow 48.5%

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

       3. inv-pow 48.5%

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

       4. pow-pow 48.5%

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

       5. metadata-eval 48.5%

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

       6. pow1/2 48.5%

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

   12. Applied egg-rr 48.5%

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

       1. unpow-1 48.5%

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

       2. associate-/r/ 48.5%

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

   14. Simplified 48.5%

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

   15. Taylor expanded in a around 0 88.8%

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

       1. +-commutative 88.8%

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

       2. mul-1-neg 88.8%

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

       3. unsub-neg 88.8%

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

   17. Simplified 88.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - {\left(\frac{1}{{b}^{2} + c \cdot \left(a \cdot -4\right)}\right)}^{-0.5}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* 4.0 (* c a))
   (- (- b) (pow (/ 1.0 (+ (pow b 2.0) (* c (* a -4.0)))) -0.5)))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((4.0 * (c * a)) / (-b - pow((1.0 / (pow(b, 2.0) + (c * (a * -4.0)))), -0.5))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((4.0d0 * (c * a)) / (-b - ((1.0d0 / ((b ** 2.0d0) + (c * (a * (-4.0d0))))) ** (-0.5d0)))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((4.0 * (c * a)) / (-b - Math.pow((1.0 / (Math.pow(b, 2.0) + (c * (a * -4.0)))), -0.5))) / (a * 2.0);
}

Python

def code(a, b, c):
	return ((4.0 * (c * a)) / (-b - math.pow((1.0 / (math.pow(b, 2.0) + (c * (a * -4.0)))), -0.5))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(4.0 * Float64(c * a)) / Float64(Float64(-b) - (Float64(1.0 / Float64((b ^ 2.0) + Float64(c * Float64(a * -4.0)))) ^ -0.5))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((4.0 * (c * a)) / (-b - ((1.0 / ((b ^ 2.0) + (c * (a * -4.0)))) ^ -0.5))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Power[N[(1.0 / N[(N[Power[b, 2.0], $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. *-commutative 54.0%

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

3. Simplified 54.0%

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

6. Applied egg-rr 53.7%

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

   1. flip-+ 53.4%

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

8. Applied egg-rr 55.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. unpow2 55.0%

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

   2. sqr-neg 55.0%

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

   3. unpow2 55.0%

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

   4. remove-double-div 55.3%

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

   5. fma-udef 55.7%

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

   6. unpow2 55.7%

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

   7. *-commutative 55.7%

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

   8. associate-*r* 55.7%

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

   9. +-commutative 55.7%

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

   10. associate-*r* 55.7%

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

   11. *-commutative 55.7%

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

   12. fma-def 55.7%

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

   13. *-commutative 55.7%

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

10. Simplified 55.7%

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

11. Taylor expanded in b around 0 99.3%

    \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}{a \cdot 2} \]
12. Step-by-step derivation

    1. fma-udef 99.3%

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

13. Applied egg-rr 99.3%

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

14. Final simplification 99.3%

    \[\leadsto \frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - {\left(\frac{1}{{b}^{2} + c \cdot \left(a \cdot -4\right)}\right)}^{-0.5}}}{a \cdot 2} \]
15. Add Preprocessing

Alternative 7: 81.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. *-commutative 54.0%

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

3. Simplified 54.0%

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

6. Applied egg-rr 53.7%

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

   1. flip-+ 53.4%

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

8. Applied egg-rr 55.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. unpow2 55.0%

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

   2. sqr-neg 55.0%

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

   3. unpow2 55.0%

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

   4. remove-double-div 55.3%

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

   5. fma-udef 55.7%

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

   6. unpow2 55.7%

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

   7. *-commutative 55.7%

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

   8. associate-*r* 55.7%

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

   9. +-commutative 55.7%

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

   10. associate-*r* 55.7%

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

   11. *-commutative 55.7%

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

   12. fma-def 55.7%

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

   13. *-commutative 55.7%

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

10. Simplified 55.7%

    \[\leadsto \frac{\color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\left(-b\right) - {\left(\frac{1}{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}\right)}^{-0.5}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. clear-num 55.7%

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

    2. inv-pow 55.7%

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

    3. inv-pow 55.7%

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

    4. pow-pow 55.7%

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

    5. metadata-eval 55.7%

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

    6. pow1/2 55.7%

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

12. Applied egg-rr 55.7%

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

    1. unpow-1 55.7%

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

    2. associate-/r/ 55.7%

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

14. Simplified 55.7%

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

15. Taylor expanded in a around 0 83.1%

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

    1. +-commutative 83.1%

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

    2. mul-1-neg 83.1%

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

    3. unsub-neg 83.1%

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

17. Simplified 83.1%

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

18. Final simplification 83.1%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
19. Add Preprocessing

Alternative 8: 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 54.0%

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

   1. *-commutative 54.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in b around inf 65.8%

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

   1. mul-1-neg 65.8%

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

   2. distribute-neg-frac 65.8%

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

7. Simplified 65.8%

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

8. Final simplification 65.8%

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

Reproduce

herbie shell --seed 2024026 
(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)))