Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.2%

Time: 12.4s

Alternatives: 6

Speedup: 2.9×

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

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

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: 55.2% 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333 \cdot \frac{c}{a}}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot \frac{0.3333333333333333}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* -0.3333333333333333 (/ c a))
  (* (+ (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))))

C

double code(double a, double b, double c) {
	return (-0.3333333333333333 * (c / a)) / ((sqrt(fma((c * -3.0), a, (b * b))) + b) * (0.3333333333333333 / a));
}

Julia

function code(a, b, c)
	return Float64(Float64(-0.3333333333333333 * Float64(c / a)) / Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b) * Float64(0.3333333333333333 / a)))
end

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 52.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

6. Applied rewrites 53.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) \cdot {\left(\frac{0.3333333333333333}{a}\right)}^{2} - {\left(\frac{b}{a \cdot 3}\right)}^{2}}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}} \]

7. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

    \[\leadsto \frac{-0.3333333333333333 \cdot \frac{c}{a}}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot \frac{0.3333333333333333}{a}} \]
11. Add Preprocessing

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.0)
   (* (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a) 0.3333333333333333)
   (/ 1.0 (fma (/ 1.5 b) a (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.0) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = 1.0 / fma((1.5 / b), a, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / fma(Float64(1.5 / b), a, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.0], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[(1.5 / b), $MachinePrecision] * a + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7

   1. Initial program 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.3%

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

   if 7 < b

   1. Initial program 46.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 46.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 88.6

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, \color{blue}{\frac{b}{c}} \cdot -2\right)} \]

   7. Applied rewrites 88.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1.5}{b}, a, \frac{b}{c} \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.0)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma (/ 1.5 b) a (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.0) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma((1.5 / b), a, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.0)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / fma(Float64(1.5 / b), a, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.0], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.5 / b), $MachinePrecision] * a + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7

   1. Initial program 79.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 79.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.3

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

   4. Applied rewrites 79.3%

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

   if 7 < b

   1. Initial program 46.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 46.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 88.6

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, \color{blue}{\frac{b}{c}} \cdot -2\right)} \]

   7. Applied rewrites 88.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1.5}{b}, a, \frac{b}{c} \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, -2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma (/ 1.5 b) a (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma((1.5 / b), a, (-2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(Float64(1.5 / b), a, Float64(-2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(1.5 / b), $MachinePrecision] * a + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 53.5

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 53.5

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 53.5

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

4. Applied rewrites 53.5%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 82.9

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, \color{blue}{\frac{b}{c}} \cdot -2\right)} \]

7. Applied rewrites 82.9%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1.5}{b}, a, \frac{b}{c} \cdot -2\right)}} \]

8. Final simplification 82.9%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1.5}{b}, a, -2 \cdot \frac{b}{c}\right)} \]
9. Add Preprocessing

Alternative 5: 64.6% 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 53.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 65.5

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

5. Applied rewrites 65.5%

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

6. Final simplification 65.5%

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

Alternative 6: 64.5% 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 53.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 65.5

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

5. Applied rewrites 65.5%

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

7. Applied rewrites 65.4%

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

8. Final simplification 65.4%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))