Cubic critical, narrow range

Percentage Accurate: 55.1% → 99.3%

Time: 17.7s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, a \cdot 3, 0\right)}{\left(a \cdot \left(-3\right)\right) \cdot \left(b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma c (* a 3.0) 0.0)
  (* (* a (- 3.0)) (+ b (sqrt (- (pow b 2.0) (* c (* a 3.0))))))))

C

double code(double a, double b, double c) {
	return fma(c, (a * 3.0), 0.0) / ((a * -3.0) * (b + sqrt((pow(b, 2.0) - (c * (a * 3.0))))));
}

Julia

function code(a, b, c)
	return Float64(fma(c, Float64(a * 3.0), 0.0) / Float64(Float64(a * Float64(-3.0)) * Float64(b + sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * 3.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * 3.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(a * (-3.0)), $MachinePrecision] * N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. expm1-log1p-u 56.9%

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

   2. expm1-undefine 55.2%

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

4. Applied egg-rr 55.2%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. add-sqr-sqrt 56.2%

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

   4. pow2 56.2%

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

   5. expm1-define 57.7%

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

   6. expm1-log1p-u 57.7%

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

   7. *-commutative 57.7%

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

   8. pow2 57.7%

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

   9. expm1-define 58.2%

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

   10. expm1-log1p-u 58.2%

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

   11. *-commutative 58.2%

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

6. Applied egg-rr 58.2%

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

   1. associate--r- 99.3%

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

   2. unpow2 99.1%

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

   3. sqr-neg 99.1%

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

   4. unpow2 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

8. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. associate-/l/ 99.3%

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

   3. +-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. fma-define 99.3%

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

   6. *-commutative 99.3%

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

   7. +-inverses 99.3%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(c, a \cdot 3, \color{blue}{0}\right)}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}\right)} \]

   8. *-commutative 99.3%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(c, a \cdot 3, 0\right)}{\color{blue}{\left(a \cdot 3\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}\right)} \]

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(c, a \cdot 3, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}\right)}} \]

11. Final simplification 99.3%

    \[\leadsto \frac{\mathsf{fma}\left(c, a \cdot 3, 0\right)}{\left(a \cdot \left(-3\right)\right) \cdot \left(b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}\right)} \]
12. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * -3.0)) / (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 = ((c * (a * -3.0d0)) / (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 ((c * (a * -3.0)) / (b + Math.sqrt((Math.pow(b, 2.0) - (c * (a * 3.0)))))) / (a * 3.0);
}

Python

def code(a, b, c):
	return ((c * (a * -3.0)) / (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(c * Float64(a * Float64(-3.0))) / 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 = ((c * (a * -3.0)) / (b + sqrt(((b ^ 2.0) - (c * (a * 3.0)))))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * (-3.0)), $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 57.0%

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

   1. expm1-log1p-u 56.9%

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

   2. expm1-undefine 55.2%

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

4. Applied egg-rr 55.2%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. add-sqr-sqrt 56.2%

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

   4. pow2 56.2%

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

   5. expm1-define 57.7%

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

   6. expm1-log1p-u 57.7%

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

   7. *-commutative 57.7%

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

   8. pow2 57.7%

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

   9. expm1-define 58.2%

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

   10. expm1-log1p-u 58.2%

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

   11. *-commutative 58.2%

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

6. Applied egg-rr 58.2%

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

   1. associate--r- 99.3%

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

   2. unpow2 99.1%

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

   3. sqr-neg 99.1%

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

   4. unpow2 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in b around 0 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 250:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 250.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 250.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 250.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 250.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 250

   1. Initial program 78.2%

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

   3. Simplified 78.3%

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

   if 250 < b

   1. Initial program 42.9%

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

   3. Taylor expanded in c around 0 88.4%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

   4. Taylor expanded in c around inf 88.3%

      \[\leadsto \frac{c \cdot \color{blue}{\left(c \cdot \left(-1.5 \cdot \frac{a}{b \cdot c} + -1.125 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}}{3 \cdot a} \]

   5. Taylor expanded in b around inf 88.9%

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

      1. fma-define 88.9%

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

      2. associate-/l* 88.9%

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

      3. unpow2 88.9%

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

      4. unpow2 88.9%

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

      5. times-frac 88.9%

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

      6. unpow2 88.9%

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

   7. Simplified 88.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 250:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 250:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 250.0)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/ (fma -0.5 c (* -0.375 (* a (pow (/ c b) 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 250.0) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, c, (-0.375 * (a * pow((c / b), 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 250.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 250.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 250

   1. Initial program 78.2%

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

   if 250 < b

   1. Initial program 42.9%

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

   3. Taylor expanded in c around 0 88.4%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

   4. Taylor expanded in c around inf 88.3%

      \[\leadsto \frac{c \cdot \color{blue}{\left(c \cdot \left(-1.5 \cdot \frac{a}{b \cdot c} + -1.125 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}}{3 \cdot a} \]

   5. Taylor expanded in b around inf 88.9%

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

      1. fma-define 88.9%

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

      2. associate-/l* 88.9%

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

      3. unpow2 88.9%

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

      4. unpow2 88.9%

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

      5. times-frac 88.9%

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

      6. unpow2 88.9%

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

   7. Simplified 88.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 250:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 250:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{{b}^{3}} + \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 250.0)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (* c (+ (/ (* -0.375 (* c a)) (pow b 3.0)) (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 250.0) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * (((-0.375 * (c * a)) / pow(b, 3.0)) + (-0.5 / b));
	}
	return tmp;
}

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) :: tmp
    if (b <= 250.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = c * ((((-0.375d0) * (c * a)) / (b ** 3.0d0)) + ((-0.5d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 250.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * (((-0.375 * (c * a)) / Math.pow(b, 3.0)) + (-0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 250.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = c * (((-0.375 * (c * a)) / math.pow(b, 3.0)) + (-0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 250.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(Float64(-0.375 * Float64(c * a)) / (b ^ 3.0)) + Float64(-0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 250.0)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = c * (((-0.375 * (c * a)) / (b ^ 3.0)) + (-0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 250.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 250

   1. Initial program 78.2%

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

   if 250 < b

   1. Initial program 42.9%

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

   3. Taylor expanded in c around 0 88.4%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

   4. Taylor expanded in c around inf 88.3%

      \[\leadsto \frac{c \cdot \color{blue}{\left(c \cdot \left(-1.5 \cdot \frac{a}{b \cdot c} + -1.125 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}}{3 \cdot a} \]

   5. Taylor expanded in c around 0 88.7%

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

      1. cancel-sign-sub-inv 88.7%

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

      2. associate-*r/ 88.7%

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

      3. metadata-eval 88.7%

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

      4. associate-*r/ 88.7%

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

      5. metadata-eval 88.7%

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

   7. Simplified 88.7%

      \[\leadsto \color{blue}{c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} + \frac{-0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 250:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{{b}^{3}} + \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (((-0.375 * (c * a)) / pow(b, 3.0)) + (-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 * ((((-0.375d0) * (c * a)) / (b ** 3.0d0)) + ((-0.5d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * (((-0.375 * (c * a)) / Math.pow(b, 3.0)) + (-0.5 / b));
}

Python

def code(a, b, c):
	return c * (((-0.375 * (c * a)) / math.pow(b, 3.0)) + (-0.5 / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((-0.375 * (c * a)) / (b ^ 3.0)) + (-0.5 / b));
end

Wolfram

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in c around 0 78.5%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]

4. Taylor expanded in c around inf 78.4%

   \[\leadsto \frac{c \cdot \color{blue}{\left(c \cdot \left(-1.5 \cdot \frac{a}{b \cdot c} + -1.125 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}}{3 \cdot a} \]

5. Taylor expanded in c around 0 78.7%

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

   1. cancel-sign-sub-inv 78.7%

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

   2. associate-*r/ 78.7%

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

   3. metadata-eval 78.7%

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

   4. associate-*r/ 78.7%

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

   5. metadata-eval 78.7%

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

7. Simplified 78.7%

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

8. Final simplification 78.7%

   \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{{b}^{3}} + \frac{-0.5}{b}\right) \]
9. Add Preprocessing

Alternative 7: 64.5% 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 57.0%

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

3. Taylor expanded in b around inf 62.9%

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

   1. associate-*r/ 62.9%

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

   2. *-commutative 62.9%

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

5. Simplified 62.9%

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

Alternative 8: 64.4% 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 57.0%

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

3. Taylor expanded in b around inf 62.9%

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

   1. associate-*r/ 62.9%

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

   2. *-commutative 62.9%

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

5. Simplified 62.9%

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

6. Taylor expanded in c around 0 62.9%

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

   1. associate-*r/ 62.9%

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

   2. *-commutative 62.9%

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

   3. associate-*r/ 62.8%

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

8. Simplified 62.8%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))