Cubic critical, medium range

Percentage Accurate: 31.5% → 99.4%

Time: 10.1s

Alternatives: 3

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.5% 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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

4. Applied rewrites 33.2%

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

6. Applied rewrites 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 90.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lower-/.f64 N/A

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

4. Applied rewrites 33.2%

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

6. Applied rewrites 33.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + \frac{3}{2} \cdot \frac{a}{b}}} \]
8. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 91.0

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

9. Applied rewrites 91.0%

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

10. Final simplification 91.0%

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

Alternative 3: 81.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 80.3

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

5. Applied rewrites 80.3%

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

6. Final simplification 80.3%

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

Reproduce

herbie shell --seed 2024283 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))