Cubic critical, wide range

Percentage Accurate: 18.2% → 99.0%

Time: 11.7s

Alternatives: 5

Speedup: 23.2×

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: 18.2% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \frac{t_0}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (* (/ t_0 (+ b (sqrt (fma b b t_0)))) (/ 1.0 (/ a 0.3333333333333333)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return (t_0 / (b + sqrt(fma(b, b, t_0)))) * (1.0 / (a / 0.3333333333333333));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	return Float64(Float64(t_0 / Float64(b + sqrt(fma(b, b, t_0)))) * Float64(1.0 / Float64(a / 0.3333333333333333)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.8%

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

   1. neg-sub0 19.8%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. associate-+l- 19.8%

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

   3. sub0-neg 19.8%

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

   4. neg-mul-1 19.8%

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

   5. associate-*r/ 19.8%

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

   6. *-commutative 19.8%

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

   7. metadata-eval 19.8%

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

   8. metadata-eval 19.8%

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

   9. times-frac 19.8%

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

   10. *-commutative 19.8%

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

   11. times-frac 19.8%

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

3. Simplified 19.9%

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

   1. clear-num 19.9%

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

   2. inv-pow 19.9%

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

5. Applied egg-rr 19.9%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 19.9%

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

7. Simplified 19.9%

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

   1. flip-- 19.7%

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

   2. add-sqr-sqrt 20.3%

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

9. Applied egg-rr 20.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} + b}} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]
10. Step-by-step derivation

    1. associate-*r* 20.3%

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

    2. *-commutative 20.3%

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

    3. associate-*l* 20.3%

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

    4. +-commutative 20.3%

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

    5. associate-*r* 20.3%

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

    6. *-commutative 20.3%

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

    7. associate-*l* 20.3%

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

11. Simplified 20.3%

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

12. Taylor expanded in b around 0 99.1%

    \[\leadsto \frac{\color{blue}{-3 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]
13. Step-by-step derivation

    1. *-commutative 99.1%

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

    2. associate-*r* 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{0.3333333333333333}} \cdot \frac{-3 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ 1.0 (/ a 0.3333333333333333))
  (/ (* -3.0 (* c a)) (+ b (sqrt (fma b b (* c (* a -3.0))))))))

C

double code(double a, double b, double c) {
	return (1.0 / (a / 0.3333333333333333)) * ((-3.0 * (c * a)) / (b + sqrt(fma(b, b, (c * (a * -3.0))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(a / 0.3333333333333333)) * Float64(Float64(-3.0 * Float64(c * a)) / Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))))))
end

Wolfram

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

Derivation

1. Initial program 19.8%

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

   1. neg-sub0 19.8%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. associate-+l- 19.8%

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

   3. sub0-neg 19.8%

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

   4. neg-mul-1 19.8%

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

   5. associate-*r/ 19.8%

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

   6. *-commutative 19.8%

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

   7. metadata-eval 19.8%

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

   8. metadata-eval 19.8%

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

   9. times-frac 19.8%

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

   10. *-commutative 19.8%

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

   11. times-frac 19.8%

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

3. Simplified 19.9%

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

   1. clear-num 19.9%

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

   2. inv-pow 19.9%

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

5. Applied egg-rr 19.9%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 19.9%

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

7. Simplified 19.9%

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

   1. flip-- 19.7%

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

   2. add-sqr-sqrt 20.3%

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

9. Applied egg-rr 20.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} + b}} \cdot \frac{1}{\frac{a}{0.3333333333333333}} \]
10. Step-by-step derivation

    1. associate-*r* 20.3%

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

    2. *-commutative 20.3%

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

    3. associate-*l* 20.3%

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

    4. +-commutative 20.3%

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

    5. associate-*r* 20.3%

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

    6. *-commutative 20.3%

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

    7. associate-*l* 20.3%

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

11. Simplified 20.3%

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

12. Taylor expanded in b around 0 99.1%

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

13. Final simplification 99.1%

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

Alternative 3: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(a \cdot -0.375\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma -0.5 (/ c b) (/ (* (* a -0.375) (* c c)) (pow b 3.0))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (((a * -0.375) * (c * c)) / pow(b, 3.0)));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(Float64(Float64(a * -0.375) * Float64(c * c)) / (b ^ 3.0)))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(a * -0.375), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.8%

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

   1. neg-sub0 19.8%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. associate-+l- 19.8%

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

   3. sub0-neg 19.8%

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

   4. neg-mul-1 19.8%

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

   5. associate-*r/ 19.8%

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

   6. metadata-eval 19.8%

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

   7. metadata-eval 19.8%

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

   8. times-frac 19.8%

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

   9. *-commutative 19.8%

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

   10. times-frac 19.8%

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

   11. associate-*l/ 19.8%

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

3. Simplified 19.9%

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

4. Taylor expanded in b around inf 94.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
5. Step-by-step derivation

   1. fma-def 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   2. associate-*r/ 94.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\frac{-0.375 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

   3. *-commutative 94.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

   4. associate-*r* 94.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\color{blue}{\left(-0.375 \cdot a\right) \cdot {c}^{2}}}{{b}^{3}}\right) \]

   5. unpow2 94.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(-0.375 \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}\right) \]

6. Simplified 94.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(-0.375 \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right)} \]

7. Final simplification 94.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\left(a \cdot -0.375\right) \cdot \left(c \cdot c\right)}{{b}^{3}}\right) \]

Alternative 4: 90.2% accurate, 23.2× 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 19.8%

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

   1. neg-sub0 19.8%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. associate-+l- 19.8%

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

   3. sub0-neg 19.8%

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

   4. neg-mul-1 19.8%

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

   5. associate-*r/ 19.8%

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

   6. metadata-eval 19.8%

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

   7. metadata-eval 19.8%

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

   8. times-frac 19.8%

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

   9. *-commutative 19.8%

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

   10. times-frac 19.8%

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

   11. associate-*l/ 19.8%

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

3. Simplified 19.9%

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

4. Taylor expanded in b around inf 89.1%

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

5. Final simplification 89.1%

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

Alternative 5: 3.3% accurate, 116.0× 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 19.8%

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

   1. neg-sub0 19.8%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. associate-+l- 19.8%

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

   3. sub0-neg 19.8%

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

   4. neg-mul-1 19.8%

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

   5. associate-*r/ 19.8%

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

   6. *-commutative 19.8%

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

   7. metadata-eval 19.8%

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

   8. metadata-eval 19.8%

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

   9. times-frac 19.8%

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

   10. *-commutative 19.8%

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

   11. times-frac 19.8%

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

3. Simplified 19.9%

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

   1. clear-num 19.9%

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

   2. inv-pow 19.9%

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

5. Applied egg-rr 19.9%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 19.9%

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

7. Simplified 19.9%

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

   1. *-un-lft-identity 19.9%

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

   2. add-sqr-sqrt 20.3%

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

   3. prod-diff 21.1%

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

   4. add-sqr-sqrt 20.9%

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

   5. add-sqr-sqrt 21.1%

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

9. Applied egg-rr 21.1%

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

10. Taylor expanded in a around 0 3.3%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
11. Step-by-step derivation

    1. associate-*r/ 3.3%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.3%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.3%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.3%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.3%

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

    6. div0 3.3%

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

12. Simplified 3.3%

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

13. Final simplification 3.3%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023249 
(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)))