Cubic critical, wide range

Percentage Accurate: 18.0% → 99.4%

Time: 16.4s

Alternatives: 7

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: 18.0% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.1%

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

   1. expm1-log1p-u 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

   2. expm1-undefine 12.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]

4. Applied egg-rr 12.9%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]
5. Step-by-step derivation

   1. expm1-define 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

6. Simplified 16.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 16.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c} \cdot \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}}{3 \cdot a} \]

   2. pow2 16.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c} \cdot \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   3. add-sqr-sqrt 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   4. pow2 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   5. expm1-log1p-u 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(3 \cdot a\right)} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   6. *-commutative 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   7. *-commutative 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   8. pow2 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   9. expm1-log1p-u 16.7%

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

   10. *-commutative 16.7%

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

   11. *-commutative 16.7%

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

8. Applied egg-rr 16.7%

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

   1. associate--r- 99.5%

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

10. Simplified 99.5%

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

    1. *-commutative 99.5%

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

    2. *-commutative 99.5%

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

    3. cancel-sign-sub-inv 99.5%

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

    4. pow2 99.5%

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

    5. fma-define 99.5%

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

    6. *-commutative 99.5%

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

12. Applied egg-rr 99.5%

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

13. Taylor expanded in b around 0 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \frac{\frac{t\_0}{\left(-b\right) - \sqrt{{b}^{2} - t\_0}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))))
   (/ (/ t_0 (- (- b) (sqrt (- (pow b 2.0) t_0)))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return (t_0 / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c * (a * 3.0d0)
    code = (t_0 / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return (t_0 / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	return (t_0 / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	return Float64(Float64(t_0 / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * 3.0);
	tmp = (t_0 / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
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[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 16.1%

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

   1. expm1-log1p-u 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

   2. expm1-undefine 12.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]

4. Applied egg-rr 12.9%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]
5. Step-by-step derivation

   1. expm1-define 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

6. Simplified 16.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 16.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c} \cdot \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}}{3 \cdot a} \]

   2. pow2 16.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c} \cdot \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   3. add-sqr-sqrt 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   4. pow2 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   5. expm1-log1p-u 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(3 \cdot a\right)} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   6. *-commutative 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   7. *-commutative 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   8. pow2 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   9. expm1-log1p-u 16.7%

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

   10. *-commutative 16.7%

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

   11. *-commutative 16.7%

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

8. Applied egg-rr 16.7%

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

   1. associate--r- 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in b around 0 99.5%

    \[\leadsto \frac{\frac{\color{blue}{0} + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{3 \cdot a} \]
12. Step-by-step derivation

    1. +-lft-identity 99.5%

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

    2. *-commutative 99.5%

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

    3. *-commutative 99.5%

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

    4. *-commutative 99.5%

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* 3.0 (* c a)) (- (- b) (sqrt (- (pow b 2.0) (* c (* a 3.0))))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return ((3.0 * (c * a)) / (-b - sqrt((pow(b, 2.0) - (c * (a * 3.0)))))) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((3.0d0 * (c * a)) / (-b - sqrt(((b ** 2.0d0) - (c * (a * 3.0d0)))))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((3.0 * (c * a)) / (-b - Math.sqrt((Math.pow(b, 2.0) - (c * (a * 3.0)))))) / (a * 3.0);
}

Python

def code(a, b, c):
	return ((3.0 * (c * a)) / (-b - math.sqrt((math.pow(b, 2.0) - (c * (a * 3.0)))))) / (a * 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * Float64(c * a)) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * 3.0)))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((3.0 * (c * a)) / (-b - sqrt(((b ^ 2.0) - (c * (a * 3.0)))))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.1%

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

   1. expm1-log1p-u 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

   2. expm1-undefine 12.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]

4. Applied egg-rr 12.9%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]
5. Step-by-step derivation

   1. expm1-define 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

6. Simplified 16.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 16.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c} \cdot \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}}{3 \cdot a} \]

   2. pow2 16.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c} \cdot \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   3. add-sqr-sqrt 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   4. pow2 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   5. expm1-log1p-u 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(3 \cdot a\right)} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   6. *-commutative 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   7. *-commutative 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   8. pow2 16.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right) \cdot c}}}{3 \cdot a} \]

   9. expm1-log1p-u 16.7%

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

   10. *-commutative 16.7%

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

   11. *-commutative 16.7%

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

8. Applied egg-rr 16.7%

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

   1. associate--r- 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in b around 0 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b))

C

double code(double a, double b, double c) {
	return fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * c + N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.1%

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

   1. expm1-log1p-u 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

   2. expm1-undefine 12.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]

4. Applied egg-rr 12.9%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot a\right)} - 1\right)} \cdot c}}{3 \cdot a} \]
5. Step-by-step derivation

   1. expm1-define 16.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

6. Simplified 16.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)} \cdot c}}{3 \cdot a} \]

7. Taylor expanded in b around inf 96.4%

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

   1. fma-define 96.4%

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

   2. associate-/l* 96.4%

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

   3. unpow2 96.4%

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

   4. unpow2 96.4%

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

   5. times-frac 96.4%

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

   6. unpow1 96.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right)\right)\right)}{b} \]

   7. pow-plus 96.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}\right)\right)}{b} \]

   8. metadata-eval 96.4%

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

9. Simplified 96.4%

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

Alternative 5: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (1.0d0 / (b * ((1.5d0 * ((c * a) / (b ** 2.0d0))) - 2.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return c * (1.0 / (b * ((1.5 * ((c * a) / Math.pow(b, 2.0))) - 2.0)));
}

Python

def code(a, b, c):
	return c * (1.0 / (b * ((1.5 * ((c * a) / math.pow(b, 2.0))) - 2.0)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * (1.0 / (b * ((1.5 * ((c * a) / (b ^ 2.0))) - 2.0)));
end

Wolfram

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

Derivation

1. Initial program 16.1%

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

3. Simplified 16.1%

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

5. Taylor expanded in c around 0 96.0%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 96.0%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   2. metadata-eval 96.0%

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

7. Simplified 96.0%

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

8. Taylor expanded in b around inf 95.9%

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

   1. clear-num 95.9%

      \[\leadsto c \cdot \color{blue}{\frac{1}{\frac{b}{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}}} \]

   2. inv-pow 95.9%

      \[\leadsto c \cdot \color{blue}{{\left(\frac{b}{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}\right)}^{-1}} \]

   3. fmm-def 95.9%

      \[\leadsto c \cdot {\left(\frac{b}{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{{b}^{2}}, -0.5\right)}}\right)}^{-1} \]

   4. associate-/l* 95.9%

      \[\leadsto c \cdot {\left(\frac{b}{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{c}{{b}^{2}}}, -0.5\right)}\right)}^{-1} \]

   5. metadata-eval 95.9%

      \[\leadsto c \cdot {\left(\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, \color{blue}{-0.5}\right)}\right)}^{-1} \]

10. Applied egg-rr 95.9%

    \[\leadsto c \cdot \color{blue}{{\left(\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, -0.5\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 95.9%

       \[\leadsto c \cdot \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, -0.5\right)}}} \]

12. Simplified 95.9%

    \[\leadsto c \cdot \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, -0.5\right)}}} \]

13. Taylor expanded in b around inf 96.1%

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

14. Final simplification 96.1%

    \[\leadsto c \cdot \frac{1}{b \cdot \left(1.5 \cdot \frac{c \cdot a}{{b}^{2}} - 2\right)} \]
15. Add Preprocessing

Alternative 6: 95.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{1}{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return c * (1.0 / ((b * -2.0) + (1.5 * ((c * a) / 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 = c * (1.0d0 / ((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))))
end function

Java

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

Python

def code(a, b, c):
	return c * (1.0 / ((b * -2.0) + (1.5 * ((c * a) / b))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * (1.0 / ((b * -2.0) + (1.5 * ((c * a) / b))));
end

Wolfram

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

Derivation

1. Initial program 16.1%

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

3. Simplified 16.1%

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

5. Taylor expanded in c around 0 96.0%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 96.0%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   2. metadata-eval 96.0%

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

7. Simplified 96.0%

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

8. Taylor expanded in b around inf 95.9%

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

   1. clear-num 95.9%

      \[\leadsto c \cdot \color{blue}{\frac{1}{\frac{b}{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}}} \]

   2. inv-pow 95.9%

      \[\leadsto c \cdot \color{blue}{{\left(\frac{b}{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}\right)}^{-1}} \]

   3. fmm-def 95.9%

      \[\leadsto c \cdot {\left(\frac{b}{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{{b}^{2}}, -0.5\right)}}\right)}^{-1} \]

   4. associate-/l* 95.9%

      \[\leadsto c \cdot {\left(\frac{b}{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{c}{{b}^{2}}}, -0.5\right)}\right)}^{-1} \]

   5. metadata-eval 95.9%

      \[\leadsto c \cdot {\left(\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, \color{blue}{-0.5}\right)}\right)}^{-1} \]

10. Applied egg-rr 95.9%

    \[\leadsto c \cdot \color{blue}{{\left(\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, -0.5\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 95.9%

       \[\leadsto c \cdot \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, -0.5\right)}}} \]

12. Simplified 95.9%

    \[\leadsto c \cdot \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{{b}^{2}}, -0.5\right)}}} \]

13. Taylor expanded in a around 0 96.1%

    \[\leadsto c \cdot \frac{1}{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}} \]

14. Final simplification 96.1%

    \[\leadsto c \cdot \frac{1}{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}} \]
15. Add Preprocessing

Alternative 7: 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 16.1%

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

3. Simplified 16.1%

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

5. Taylor expanded in b around inf 91.6%

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

Reproduce

herbie shell --seed 2024154 
(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)))