Cubic critical, medium range

Percentage Accurate: 31.5% → 99.3%

Time: 15.1s

Alternatives: 9

Speedup: 23.2×

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: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((pow(-b, 2.0) - pow(b, 2.0)) + (c * (a * 3.0))) / (-b - sqrt((pow(b, 2.0) + (-3.0 * (c * a)))))) / (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 = ((((-b ** 2.0d0) - (b ** 2.0d0)) + (c * (a * 3.0d0))) / (-b - sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. log1p-expm1-u 21.4%

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

   2. log1p-udef 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   3. associate-*l* 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 14.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}}{3 \cdot a} \]

   2. pow2 14.9%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   3. add-sqr-sqrt 15.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   4. pow2 15.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   5. log1p-def 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   6. log1p-expm1-u 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   7. pow2 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   8. log1p-def 21.9%

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

   9. log1p-expm1-u 35.9%

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

6. Applied egg-rr 35.9%

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

   1. associate--r- 99.1%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

8. Simplified 99.1%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.1%

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

   4. add-log-exp 39.1%

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

   5. associate-*r* 39.1%

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

   6. exp-prod 24.2%

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

   7. exp-prod 24.2%

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

10. Applied egg-rr 24.2%

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

    1. log-pow 46.7%

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

    2. log-pow 99.3%

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

    3. rem-log-exp 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((pow(-b, 2.0) - pow(b, 2.0)) + (3.0 * (c * a))) / (-b - sqrt((pow(b, 2.0) + (-3.0 * (c * a)))))) / (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 = ((((-b ** 2.0d0) - (b ** 2.0d0)) + (3.0d0 * (c * a))) / (-b - sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. log1p-expm1-u 21.4%

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

   2. log1p-udef 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   3. associate-*l* 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 14.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}}{3 \cdot a} \]

   2. pow2 14.9%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   3. add-sqr-sqrt 15.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   4. pow2 15.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   5. log1p-def 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   6. log1p-expm1-u 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   7. pow2 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   8. log1p-def 21.9%

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

   9. log1p-expm1-u 35.9%

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

6. Applied egg-rr 35.9%

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

   1. associate--r- 99.1%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in a around 0 99.1%

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

    1. *-commutative 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((pow(-b, 2.0) - pow(b, 2.0)) + (a * (c * 3.0))) / (-b - sqrt((pow(b, 2.0) + (-3.0 * (c * a)))))) / (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 = ((((-b ** 2.0d0) - (b ** 2.0d0)) + (a * (c * 3.0d0))) / (-b - sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. log1p-expm1-u 21.4%

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

   2. log1p-udef 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   3. associate-*l* 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 14.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}}{3 \cdot a} \]

   2. pow2 14.9%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)} \cdot \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   3. add-sqr-sqrt 15.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   4. pow2 15.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   5. log1p-def 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   6. log1p-expm1-u 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   7. pow2 19.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   8. log1p-def 21.9%

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

   9. log1p-expm1-u 35.9%

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

6. Applied egg-rr 35.9%

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

   1. associate--r- 99.1%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 4: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. log1p-expm1-u 21.4%

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

   2. log1p-udef 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   3. associate-*l* 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 15.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}} \]

   2. inv-pow 15.1%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1}} \]

   3. *-commutative 15.1%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1} \]

   4. neg-mul-1 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1} \]

   5. fma-def 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}}\right)}^{-1} \]

   6. pow2 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}^{-1} \]

   7. log1p-def 21.4%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}\right)}^{-1} \]

   8. log1p-expm1-u 35.2%

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

6. Applied egg-rr 35.2%

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

7. Taylor expanded in b around inf 88.5%

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

8. Final simplification 88.5%

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

Alternative 5: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. log1p-expm1-u 21.4%

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

   2. log1p-udef 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   3. associate-*l* 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 15.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}} \]

   2. inv-pow 15.1%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1}} \]

   3. *-commutative 15.1%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1} \]

   4. neg-mul-1 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1} \]

   5. fma-def 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}}\right)}^{-1} \]

   6. pow2 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}^{-1} \]

   7. log1p-def 21.4%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}\right)}^{-1} \]

   8. log1p-expm1-u 35.2%

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

6. Applied egg-rr 35.2%

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

7. Taylor expanded in b around inf 88.5%

   \[\leadsto {\color{blue}{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}}^{-1} \]
8. Step-by-step derivation

   1. expm1-log1p-u 78.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\right)\right)} \]

   2. expm1-udef 37.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\right)} - 1} \]

   3. unpow-1 37.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}}\right)} - 1 \]

   4. fma-def 37.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}}\right)} - 1 \]

   5. associate-*r/ 37.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \color{blue}{\frac{1.5 \cdot a}{b}}\right)}\right)} - 1 \]

9. Applied egg-rr 37.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5 \cdot a}{b}\right)}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 78.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5 \cdot a}{b}\right)}\right)\right)} \]

    2. expm1-log1p 88.5%

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

    3. *-commutative 88.5%

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

11. Simplified 88.5%

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

12. Final simplification 88.5%

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

Alternative 6: 81.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in b around inf 78.1%

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

   1. associate-*r/ 78.1%

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

   2. associate-/l* 77.9%

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

5. Simplified 77.9%

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

   1. associate-/r/ 77.9%

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

7. Applied egg-rr 77.9%

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

8. Final simplification 77.9%

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

Alternative 7: 81.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in b around inf 78.1%

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

   1. associate-*r/ 78.1%

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

   2. associate-/l* 77.9%

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

5. Simplified 77.9%

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

6. Final simplification 77.9%

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

Alternative 8: 81.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Taylor expanded in b around inf 78.1%

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

   1. associate-*r/ 78.1%

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

5. Simplified 78.1%

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

6. Final simplification 78.1%

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

Alternative 9: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

   1. log1p-expm1-u 21.4%

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

   2. log1p-udef 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   3. associate-*l* 15.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right)}}{3 \cdot a} \]

4. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 15.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}} \]

   2. inv-pow 15.1%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1}} \]

   3. *-commutative 15.1%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1} \]

   4. neg-mul-1 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}^{-1} \]

   5. fma-def 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}}\right)}^{-1} \]

   6. pow2 15.1%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - \log \left(1 + \mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}^{-1} \]

   7. log1p-def 21.4%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}\right)}\right)}^{-1} \]

   8. log1p-expm1-u 35.2%

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

6. Applied egg-rr 35.2%

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

7. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))