Cubic critical, wide range

Percentage Accurate: 17.6% → 97.8%

Time: 12.9s

Alternatives: 11

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Initial Program: 17.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{t\_0 \cdot \left(b \cdot b\right)}, -0.5625 \cdot \left(c \cdot c\right), \left(\frac{6.328125}{\left(t\_0 \cdot b\right) \cdot t\_0} \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot \left(-0.16666666666666666 \cdot a\right)\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{t\_0}\right), a, -0.5 \cdot \frac{c}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (fma
    (fma
     (fma
      (/ c (* t_0 (* b b)))
      (* -0.5625 (* c c))
      (*
       (* (/ 6.328125 (* (* t_0 b) t_0)) (* (* (* c c) c) c))
       (* -0.16666666666666666 a)))
     a
     (/ (* -0.375 (* c c)) t_0))
    a
    (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma(fma(fma((c / (t_0 * (b * b))), (-0.5625 * (c * c)), (((6.328125 / ((t_0 * b) * t_0)) * (((c * c) * c) * c)) * (-0.16666666666666666 * a))), a, ((-0.375 * (c * c)) / t_0)), a, (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return fma(fma(fma(Float64(c / Float64(t_0 * Float64(b * b))), Float64(-0.5625 * Float64(c * c)), Float64(Float64(Float64(6.328125 / Float64(Float64(t_0 * b) * t_0)) * Float64(Float64(Float64(c * c) * c) * c)) * Float64(-0.16666666666666666 * a))), a, Float64(Float64(-0.375 * Float64(c * c)) / t_0)), a, Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(c / N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5625 * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(6.328125 / N[(N[(t$95$0 * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \left(\left(c \cdot c\right) \cdot \frac{c}{{b}^{5}}\right) \cdot -0.5625\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

7. Applied rewrites 97.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}, \left(c \cdot c\right) \cdot -0.5625, \left(a \cdot -0.16666666666666666\right) \cdot \left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right) \cdot \frac{6.328125}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}\right)\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right) \]

8. Final simplification 97.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625 \cdot \left(c \cdot c\right), \left(\frac{6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot \left(-0.16666666666666666 \cdot a\right)\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}\right), a, -0.5 \cdot \frac{c}{b}\right) \]
9. Add Preprocessing

Alternative 2: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.2)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.2) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.20000000000000001

   1. Initial program 71.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}{3 \cdot a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c}\right)}}{3 \cdot a} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c}\right)}}{3 \cdot a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c\right)}}{3 \cdot a} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot c\right)}}{3 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot c\right)}}{3 \cdot a} \]

      11. metadata-eval 71.6

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot a\right) \cdot c\right)}}{3 \cdot a} \]

   4. Applied rewrites 71.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

   if -0.20000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 12.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.6

         \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.2)
   (/ (- (sqrt (fma b b (* (* -3.0 c) a))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.2) {
		tmp = (sqrt(fma(b, b, ((-3.0 * c) * a))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * c) * a))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.20000000000000001

   1. Initial program 71.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 71.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}} - b}{a \cdot 3} \]

      5. lower-*.f64 71.6

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot c\right) \cdot a}\right)} - b}{a \cdot 3} \]

   6. Applied rewrites 71.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}} - b}{a \cdot 3} \]

   if -0.20000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 12.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.6

         \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.2)
   (/ (* (- (sqrt (fma (* -3.0 c) a (* b b))) b) 0.3333333333333333) a)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.2) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.2)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.20000000000000001

   1. Initial program 71.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{3 \cdot a}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}} \]

   if -0.20000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 12.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.6

         \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.2)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.2) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.2)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.20000000000000001

   1. Initial program 71.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{3 \cdot a}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}}{3}} \]

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333} \]

   if -0.20000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 12.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.6

         \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{c}{b \cdot b} \cdot a, -0.5625, -0.375\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, a, -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ (* (fma (* (/ c (* b b)) a) -0.5625 -0.375) (* c c)) (* (* b b) b))
  a
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma(((fma(((c / (b * b)) * a), -0.5625, -0.375) * (c * c)) / ((b * b) * b)), a, (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(fma(Float64(Float64(c / Float64(b * b)) * a), -0.5625, -0.375) * Float64(c * c)) / Float64(Float64(b * b) * b)), a, Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.5625 + -0.375), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \left(\left(c \cdot c\right) \cdot \frac{c}{{b}^{5}}\right) \cdot -0.5625\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{\left(b \cdot b\right) \cdot b}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{c}{b} \cdot -0.5\right) \]

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{3}{8}\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 96.7%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -0.5625, -0.375\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{c}{b} \cdot -0.5\right) \]

12. Final simplification 96.7%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{c}{b \cdot b} \cdot a, -0.5625, -0.375\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, a, -0.5 \cdot \frac{c}{b}\right) \]
13. Add Preprocessing

Alternative 7: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c}{b \cdot b} \cdot c, -0.5 \cdot c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (* -0.375 a) (* (/ c (* b b)) c) (* -0.5 c)) b))

C

double code(double a, double b, double c) {
	return fma((-0.375 * a), ((c / (b * b)) * c), (-0.5 * c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-0.375 * a), Float64(Float64(c / Float64(b * b)) * c), Float64(-0.5 * c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-3}{8} \cdot a}, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   13. lower-*.f64 95.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, \color{blue}{-0.5 \cdot c}\right)}{b} \]

5. Applied rewrites 95.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right)}{b}} \]

6. Final simplification 95.1%

   \[\leadsto \frac{\mathsf{fma}\left(-0.375 \cdot a, \frac{c}{b \cdot b} \cdot c, -0.5 \cdot c\right)}{b} \]
7. Add Preprocessing

Alternative 8: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.375 \cdot c, \frac{a}{\left(b \cdot b\right) \cdot b}, \frac{-0.5}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (fma (* -0.375 c) (/ a (* (* b b) b)) (/ -0.5 b)) c))

C

double code(double a, double b, double c) {
	return fma((-0.375 * c), (a / ((b * b) * b)), (-0.5 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(-0.375 * c), Float64(a / Float64(Float64(b * b) * b)), Float64(-0.5 / b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.375 * c), $MachinePrecision] * N[(a / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \cdot c \]

   3. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \cdot c \]

   4. associate-*r* N/A

      \[\leadsto \left(\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \cdot c \]

   5. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \cdot c \]

   6. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \cdot c \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \cdot c} \]

5. Applied rewrites 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot -0.375, \frac{a}{\left(b \cdot b\right) \cdot b}, \frac{-0.5}{b}\right) \cdot c} \]

6. Final simplification 94.8%

   \[\leadsto \mathsf{fma}\left(-0.375 \cdot c, \frac{a}{\left(b \cdot b\right) \cdot b}, \frac{-0.5}{b}\right) \cdot c \]
7. Add Preprocessing

Alternative 9: 90.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 90.2

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

5. Applied rewrites 90.2%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

6. Final simplification 90.2%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]
7. Add Preprocessing

Alternative 10: 90.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{b} \cdot c \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 90.2

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

5. Applied rewrites 90.2%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 89.9%

   \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b}} \]

8. Final simplification 89.9%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Alternative 11: 3.3% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 18.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 18.5%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)\right)}}{3} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)}{3} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{1}{a} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}}{3} \]

   6. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(\mathsf{neg}\left(b\right)\right) + \frac{1}{a} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}}{3} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{a}, \mathsf{neg}\left(b\right), \frac{1}{a} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}}{3} \]

   8. lower-*.f64 20.0

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

6. Applied rewrites 20.0%

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

7. Taylor expanded in c around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \left(\color{blue}{0} \cdot \frac{b}{a}\right) \]

   3. mul0-lft N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{0} \]

   4. metadata-eval 3.3

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))