Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.3%

Time: 13.6s

Alternatives: 9

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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{1}{c} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -1.0 (* (/ 1.0 c) (+ b (sqrt (fma c (* a -3.0) (* b b)))))))

C

double code(double a, double b, double c) {
	return -1.0 / ((1.0 / c) * (b + sqrt(fma(c, (a * -3.0), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / Float64(Float64(1.0 / c) * Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))))
end

Wolfram

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 55.4%

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

6. Applied rewrites 57.2%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, b \cdot 2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.2)
   (* (- b (sqrt (fma b b (* c (* a -3.0))))) (/ 1.0 (* a -3.0)))
   (/ -1.0 (/ (fma -1.5 (/ (* a c) b) (* b 2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.2) {
		tmp = (b - sqrt(fma(b, b, (c * (a * -3.0))))) * (1.0 / (a * -3.0));
	} else {
		tmp = -1.0 / (fma(-1.5, ((a * c) / b), (b * 2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.2)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) * Float64(1.0 / Float64(a * -3.0)));
	else
		tmp = Float64(-1.0 / Float64(fma(-1.5, Float64(Float64(a * c) / b), Float64(b * 2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.2], N[(N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.199999999999999

   1. Initial program 81.4%

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

   4. Applied rewrites 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 81.6%

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

   if 15.199999999999999 < b

   1. Initial program 47.4%

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, b \cdot 2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, b \cdot 2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.2)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/ -1.0 (/ (fma -1.5 (/ (* a c) b) (* b 2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -1.0 / (fma(-1.5, ((a * c) / b), (b * 2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-1.0 / Float64(fma(-1.5, Float64(Float64(a * c) / b), Float64(b * 2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.2], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.199999999999999

   1. Initial program 81.4%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 81.5%

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

   if 15.199999999999999 < b

   1. Initial program 47.4%

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, b \cdot 2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(2, \frac{b}{c}, -1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.2)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/ -1.0 (fma 2.0 (/ b c) (* -1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -1.0 / fma(2.0, (b / c), (-1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-1.0 / fma(2.0, Float64(b / c), Float64(-1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.2], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(2.0 * N[(b / c), $MachinePrecision] + N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.199999999999999

   1. Initial program 81.4%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 81.5%

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

   if 15.199999999999999 < b

   1. Initial program 47.4%

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

Alternative 5: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(2, \frac{b}{c}, -1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.2)
   (/ (* (- (sqrt (fma b b (* c (* a -3.0)))) b) 0.3333333333333333) a)
   (/ -1.0 (fma 2.0 (/ b c) (* -1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.2) {
		tmp = ((sqrt(fma(b, b, (c * (a * -3.0)))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -1.0 / fma(2.0, (b / c), (-1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.2)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-1.0 / fma(2.0, Float64(b / c), Float64(-1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.2], N[(N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 / N[(2.0 * N[(b / c), $MachinePrecision] + N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.199999999999999

   1. Initial program 81.4%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 81.5%

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

   if 15.199999999999999 < b

   1. Initial program 47.4%

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(2, \frac{b}{c}, -1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.2)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) a))
   (/ -1.0 (fma 2.0 (/ b c) (* -1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.2) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (c * (a * -3.0)))) - b) / a);
	} else {
		tmp = -1.0 / fma(2.0, (b / c), (-1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.2)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / a));
	else
		tmp = Float64(-1.0 / fma(2.0, Float64(b / c), Float64(-1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.2], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(2.0 * N[(b / c), $MachinePrecision] + N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.199999999999999

   1. Initial program 81.4%

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

   4. Applied rewrites 81.4%

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

   6. Applied rewrites 81.5%

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

   if 15.199999999999999 < b

   1. Initial program 47.4%

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.2:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(2, \frac{b}{c}, -1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(2, \frac{b}{c}, -1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.5)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (/ -1.0 (fma 2.0 (/ b c) (* -1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = -1.0 / fma(2.0, (b / c), (-1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(-1.0 / fma(2.0, Float64(b / c), Float64(-1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.5], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(2.0 * N[(b / c), $MachinePrecision] + N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 81.7%

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

   4. Applied rewrites 81.7%

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

   if 10.5 < b

   1. Initial program 47.7%

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(2, \frac{b}{c}, -1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 55.4%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 81.5

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

7. Applied rewrites 81.5%

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

Alternative 9: 64.5% 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 55.4%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.1

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

5. Applied rewrites 64.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))