Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.5%

Time: 27.5s

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}}\\ \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{\frac{\frac{c}{b}}{\frac{b}{c}}}{b}, \frac{c}{b}\right), {a}^{3} \cdot \left(\frac{\mathsf{fma}\left(16, t\_0, 4 \cdot t\_0\right)}{b} \cdot -0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow c 4.0) (pow b 6.0))))
   (if (<= b 1.1)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (fma
      -2.0
      (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0))
      (fma
       -1.0
       (fma a (/ (/ (/ c b) (/ b c)) b) (/ c b))
       (* (pow a 3.0) (* (/ (fma 16.0 t_0 (* 4.0 t_0)) b) -0.25)))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(c, 4.0) / pow(b, 6.0);
	double tmp;
	if (b <= 1.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)), fma(-1.0, fma(a, (((c / b) / (b / c)) / b), (c / b)), (pow(a, 3.0) * ((fma(16.0, t_0, (4.0 * t_0)) / b) * -0.25))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((c ^ 4.0) / (b ^ 6.0))
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0)), fma(-1.0, fma(a, Float64(Float64(Float64(c / b) / Float64(b / c)) / b), Float64(c / b)), Float64((a ^ 3.0) * Float64(Float64(fma(16.0, t_0, Float64(4.0 * t_0)) / b) * -0.25))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.1], 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[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a * N[(N[(N[(c / b), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(N[(16.0 * t$95$0 + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. +-commutative 85.9%

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

      3. sqr-neg 85.9%

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

      4. unsub-neg 85.9%

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

      5. sqr-neg 85.9%

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

      6. fma-neg 85.9%

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

      7. distribute-lft-neg-in 85.9%

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

      8. *-commutative 85.9%

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

      9. *-commutative 85.9%

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

      10. distribute-rgt-neg-in 85.9%

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

      11. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

   if 1.1000000000000001 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in a around 0 93.2%

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

   7. Simplified 93.2%

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

      1. unpow2 93.2%

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

      2. unpow3 93.2%

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

      3. times-frac 93.2%

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

      4. pow2 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. associate-*r/ 93.2%

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

       2. associate-*l/ 93.2%

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

       3. unpow2 93.2%

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

       4. frac-times 93.2%

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

       5. pow2 93.2%

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

   11. Applied egg-rr 93.2%

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

       1. unpow2 93.2%

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

       2. clear-num 93.2%

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

       3. un-div-inv 93.2%

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

   13. Applied egg-rr 93.2%

       \[\leadsto \mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{\color{blue}{\frac{\frac{c}{b}}{\frac{b}{c}}}}{b}, \frac{c}{b}\right), {a}^{3} \cdot \left(\frac{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b} \cdot -0.25\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{\frac{\frac{c}{b}}{\frac{b}{c}}}{b}, \frac{c}{b}\right), {a}^{3} \cdot \left(\frac{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b} \cdot -0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({c}^{4} \cdot {a}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (+
    (* -2.0 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
    (-
     (-
      (*
       -0.25
       (/
        (+
         (* 16.0 (* (pow c 4.0) (pow a 4.0)))
         (pow (* -2.0 (* (pow a 2.0) (pow c 2.0))) 2.0))
        (* a (pow b 7.0))))
      (/ (* a (pow c 2.0)) (pow b 3.0)))
     (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) + (((-0.25 * (((16.0 * (pow(c, 4.0) * pow(a, 4.0))) + pow((-2.0 * (pow(a, 2.0) * pow(c, 2.0))), 2.0)) / (a * pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(16.0 * Float64((c ^ 4.0) * (a ^ 4.0))) + (Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 2.0))) ^ 2.0)) / Float64(a * (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], 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[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. +-commutative 85.9%

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

      3. sqr-neg 85.9%

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

      4. unsub-neg 85.9%

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

      5. sqr-neg 85.9%

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

      6. fma-neg 85.9%

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

      7. distribute-lft-neg-in 85.9%

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

      8. *-commutative 85.9%

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

      9. *-commutative 85.9%

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

      10. distribute-rgt-neg-in 85.9%

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

      11. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

   if 1.1000000000000001 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf 93.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({c}^{4} \cdot {a}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (* -2.0 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
    (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], 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[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. +-commutative 85.9%

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

      3. sqr-neg 85.9%

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

      4. unsub-neg 85.9%

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

      5. sqr-neg 85.9%

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

      6. fma-neg 85.9%

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

      7. distribute-lft-neg-in 85.9%

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

      8. *-commutative 85.9%

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

      9. *-commutative 85.9%

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

      10. distribute-rgt-neg-in 85.9%

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

      11. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

   if 1.1000000000000001 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf 91.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a}{b}\\ \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot t\_0 + -2 \cdot \left(\frac{c \cdot a}{{b}^{2}} \cdot t\_0\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c a) b)))
   (if (<= b 1.1)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (/
      (+
       (* -4.0 (/ (* (pow c 3.0) (pow a 3.0)) (pow b 5.0)))
       (+ (* -2.0 t_0) (* -2.0 (* (/ (* c a) (pow b 2.0)) t_0))))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) / b;
	double tmp;
	if (b <= 1.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-4.0 * ((pow(c, 3.0) * pow(a, 3.0)) / pow(b, 5.0))) + ((-2.0 * t_0) + (-2.0 * (((c * a) / pow(b, 2.0)) * t_0)))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) / b)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-4.0 * Float64(Float64((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-2.0 * t_0) + Float64(-2.0 * Float64(Float64(Float64(c * a) / (b ^ 2.0)) * t_0)))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 1.1], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(-2.0 * N[(N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. +-commutative 85.9%

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

      3. sqr-neg 85.9%

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

      4. unsub-neg 85.9%

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

      5. sqr-neg 85.9%

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

      6. fma-neg 85.9%

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

      7. distribute-lft-neg-in 85.9%

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

      8. *-commutative 85.9%

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

      9. *-commutative 85.9%

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

      10. distribute-rgt-neg-in 85.9%

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

      11. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

   if 1.1000000000000001 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around inf 90.8%

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

      1. pow-prod-down 90.8%

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

      2. rem-cbrt-cube 90.8%

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

      3. rem-cbrt-cube 90.8%

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

      4. cbrt-prod 90.8%

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

      5. pow2 90.8%

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

      6. unpow3 90.8%

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

      7. times-frac 90.8%

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

      8. *-commutative 90.8%

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

      9. cbrt-prod 90.8%

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

      10. rem-cbrt-cube 90.8%

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

      11. rem-cbrt-cube 90.8%

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

      12. pow2 90.8%

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

      13. *-commutative 90.8%

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

      14. cbrt-prod 90.8%

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

      15. rem-cbrt-cube 90.8%

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

      16. rem-cbrt-cube 90.8%

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

   7. Applied egg-rr 90.8%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \color{blue}{\left(\frac{c \cdot a}{{b}^{2}} \cdot \frac{c \cdot a}{b}\right)}\right)}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{c \cdot a}{b} + -2 \cdot \left(\frac{c \cdot a}{{b}^{2}} \cdot \frac{c \cdot a}{b}\right)\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 48:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 48.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (+ c (* a (pow (/ c b) 2.0))) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 48.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c + (a * pow((c / b), 2.0))) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 48.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / b) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 48.0], 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[(N[(c + N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 48

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. +-commutative 80.7%

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

      3. sqr-neg 80.7%

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

      4. unsub-neg 80.7%

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

      5. sqr-neg 80.7%

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

      6. fma-neg 80.8%

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

      7. distribute-lft-neg-in 80.8%

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

      8. *-commutative 80.8%

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

      9. *-commutative 80.8%

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

      10. distribute-rgt-neg-in 80.8%

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

      11. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

   if 48 < b

   1. Initial program 46.5%

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

      1. *-commutative 46.5%

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

   3. Simplified 46.5%

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

   5. Taylor expanded in b around inf 88.9%

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

      1. distribute-lft-out 88.9%

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

      2. associate-/l* 88.9%

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

   7. Simplified 88.9%

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

      1. unpow2 95.1%

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

      2. unpow3 95.1%

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

      3. times-frac 95.1%

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

      4. pow2 95.1%

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

   9. Applied egg-rr 88.9%

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

   10. Taylor expanded in c around 0 88.9%

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

   12. Simplified 88.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 48:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 48:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 48.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (+ c (* a (pow (/ c b) 2.0))) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 48.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c + (a * pow((c / b), 2.0))) / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 48.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (c + (a * ((c / b) ** 2.0d0))) / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 48.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c + (a * Math.pow((c / b), 2.0))) / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 48.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (c + (a * math.pow((c / b), 2.0))) / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 48.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c + Float64(a * (Float64(c / b) ^ 2.0))) / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 48.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (c + (a * ((c / b) ^ 2.0))) / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 48.0], 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], N[(N[(c + N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 48

   1. Initial program 80.7%

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

   if 48 < b

   1. Initial program 46.5%

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

      1. *-commutative 46.5%

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

   3. Simplified 46.5%

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

   5. Taylor expanded in b around inf 88.9%

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

      1. distribute-lft-out 88.9%

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

      2. associate-/l* 88.9%

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

   7. Simplified 88.9%

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

      1. unpow2 95.1%

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

      2. unpow3 95.1%

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

      3. times-frac 95.1%

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

      4. pow2 95.1%

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

   9. Applied egg-rr 88.9%

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

   10. Taylor expanded in c around 0 88.9%

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

   12. Simplified 88.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 48:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot {\left(\frac{c}{b}\right)}^{2}}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 80.8%

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

   1. distribute-lft-out 80.8%

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

   2. associate-/l* 80.8%

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

7. Simplified 80.8%

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

   1. unpow2 89.3%

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

   2. unpow3 89.3%

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

   3. times-frac 89.3%

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

   4. pow2 89.3%

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

9. Applied egg-rr 80.8%

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

    1. *-un-lft-identity 80.8%

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

    2. unpow2 80.8%

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

    3. times-frac 80.8%

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

11. Applied egg-rr 80.8%

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

12. Final simplification 80.8%

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

Alternative 8: 64.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 63.9%

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

   1. mul-1-neg 63.9%

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

   2. distribute-neg-frac 63.9%

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

7. Simplified 63.9%

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

8. Final simplification 63.9%

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

Reproduce

herbie shell --seed 2024045 
(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)))