Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.3%

Time: 12.3s

Alternatives: 7

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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

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

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 * -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 56.1%

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

   1. neg-sub0 56.1%

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

   2. associate-+l- 56.1%

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

   3. sub0-neg 56.1%

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

   4. neg-mul-1 56.1%

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

   5. associate-*r/ 56.1%

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

   6. metadata-eval 56.1%

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

   7. metadata-eval 56.1%

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

   8. times-frac 56.1%

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

   9. *-commutative 56.1%

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

   10. times-frac 56.1%

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

   11. associate-*l/ 56.1%

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

3. Simplified 56.1%

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

   1. flip-+ 55.9%

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

   2. pow2 55.9%

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

   3. add-sqr-sqrt 57.4%

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

   4. cancel-sign-sub-inv 57.4%

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

   5. fma-def 57.2%

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

   6. metadata-eval 57.2%

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

   7. *-commutative 57.2%

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

   8. cancel-sign-sub-inv 57.2%

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

   9. fma-def 57.2%

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

   10. metadata-eval 57.2%

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

   11. *-commutative 57.2%

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

5. Applied egg-rr 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*r* 57.2%

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

   3. *-commutative 57.2%

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

   4. associate-*r* 57.2%

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

7. Simplified 57.2%

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

8. Taylor expanded in b around 0 99.3%

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

   1. pow1 99.3%

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

10. Applied egg-rr 99.3%

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

    1. unpow1 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. neg-sub0 56.1%

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

   2. associate-+l- 56.1%

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

   3. sub0-neg 56.1%

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

   4. neg-mul-1 56.1%

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

   5. associate-*r/ 56.1%

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

   6. metadata-eval 56.1%

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

   7. metadata-eval 56.1%

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

   8. times-frac 56.1%

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

   9. *-commutative 56.1%

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

   10. times-frac 56.1%

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

   11. associate-*l/ 56.1%

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

3. Simplified 56.1%

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

   1. flip-+ 55.9%

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

   2. pow2 55.9%

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

   3. add-sqr-sqrt 57.4%

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

   4. cancel-sign-sub-inv 57.4%

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

   5. fma-def 57.2%

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

   6. metadata-eval 57.2%

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

   7. *-commutative 57.2%

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

   8. cancel-sign-sub-inv 57.2%

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

   9. fma-def 57.2%

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

   10. metadata-eval 57.2%

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

   11. *-commutative 57.2%

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

5. Applied egg-rr 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-*r* 57.2%

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

   3. *-commutative 57.2%

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

   4. associate-*r* 57.2%

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

7. Simplified 57.2%

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

8. Taylor expanded in b around 0 99.3%

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

   1. div-inv 99.1%

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

10. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. associate-*l* 99.1%

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

    4. *-lft-identity 99.1%

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

    5. times-frac 99.3%

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

    6. metadata-eval 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (+ (* -0.5 (/ c b)) (/ (* a -0.375) (/ (pow b 3.0) (* c c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = (-0.5 * (c / b)) + ((a * -0.375) / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(a * -0.375) / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5

   1. Initial program 79.0%

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

      1. /-rgt-identity 79.0%

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

      2. metadata-eval 79.0%

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

      3. associate-/l* 79.0%

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

      4. associate-*r/ 79.0%

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

      5. *-commutative 79.0%

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

      6. associate-*l/ 79.0%

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

      7. associate-*r/ 79.0%

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

      8. metadata-eval 79.0%

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

      9. metadata-eval 79.0%

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

      10. times-frac 79.0%

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

      11. neg-mul-1 79.0%

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

      12. distribute-rgt-neg-in 79.0%

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

      13. times-frac 79.0%

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

      14. metadata-eval 79.0%

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

      15. neg-mul-1 79.0%

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

   3. Simplified 79.1%

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

   if 9.5 < b

   1. Initial program 48.8%

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

      1. neg-sub0 48.8%

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

      2. associate-+l- 48.8%

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

      3. sub0-neg 48.8%

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

      4. neg-mul-1 48.8%

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

      5. associate-*r/ 48.8%

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

      6. metadata-eval 48.8%

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

      7. metadata-eval 48.8%

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

      8. times-frac 48.8%

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

      9. *-commutative 48.8%

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

      10. times-frac 48.8%

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

      11. associate-*l/ 48.8%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in b around inf 88.1%

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

      1. fma-def 88.1%

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

      2. associate-*r/ 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*r* 88.1%

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

      5. unpow2 88.1%

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

   6. Simplified 88.1%

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

      1. fma-udef 88.1%

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

      2. associate-/l* 88.1%

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

      3. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ 0.3333333333333333 a))
   (+ (* -0.5 (/ c b)) (/ (* a -0.375) (/ (pow b 3.0) (* c c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (-0.5 * (c / b)) + ((a * -0.375) / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(a * -0.375) / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5

   1. Initial program 79.0%

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

      1. neg-sub0 79.0%

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

      2. associate-+l- 79.0%

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

      3. sub0-neg 79.0%

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

      4. neg-mul-1 79.0%

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

      5. associate-*r/ 79.0%

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

      6. *-commutative 79.0%

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

      7. metadata-eval 79.0%

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

      8. metadata-eval 79.0%

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

      9. times-frac 79.0%

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

      10. *-commutative 79.0%

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

      11. times-frac 79.0%

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

   3. Simplified 79.1%

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

   if 9.5 < b

   1. Initial program 48.8%

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

      1. neg-sub0 48.8%

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

      2. associate-+l- 48.8%

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

      3. sub0-neg 48.8%

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

      4. neg-mul-1 48.8%

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

      5. associate-*r/ 48.8%

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

      6. metadata-eval 48.8%

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

      7. metadata-eval 48.8%

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

      8. times-frac 48.8%

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

      9. *-commutative 48.8%

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

      10. times-frac 48.8%

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

      11. associate-*l/ 48.8%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in b around inf 88.1%

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

      1. fma-def 88.1%

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

      2. associate-*r/ 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*r* 88.1%

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

      5. unpow2 88.1%

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

   6. Simplified 88.1%

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

      1. fma-udef 88.1%

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

      2. associate-/l* 88.1%

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

      3. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 5: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (/ (* a -0.375) (/ (pow b 3.0) (* c c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + ((a * -0.375) / (pow(b, 3.0) / (c * c)));
	}
	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 <= 9.5d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + ((a * (-0.375d0)) / ((b ** 3.0d0) / (c * c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + ((a * -0.375) / (Math.pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.5:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + ((a * -0.375) / (math.pow(b, 3.0) / (c * c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(a * -0.375) / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.5)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + ((a * -0.375) / ((b ^ 3.0) / (c * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5

   1. Initial program 79.0%

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

      1. neg-sub0 79.0%

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

      2. associate-+l- 79.0%

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

      3. sub0-neg 79.0%

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

      4. neg-mul-1 79.0%

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

      5. associate-*r/ 79.0%

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

      6. metadata-eval 79.0%

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

      7. metadata-eval 79.0%

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

      8. times-frac 79.0%

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

      9. *-commutative 79.0%

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

      10. times-frac 79.0%

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

      11. associate-*l/ 79.0%

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

   3. Simplified 79.0%

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

   if 9.5 < b

   1. Initial program 48.8%

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

      1. neg-sub0 48.8%

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

      2. associate-+l- 48.8%

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

      3. sub0-neg 48.8%

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

      4. neg-mul-1 48.8%

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

      5. associate-*r/ 48.8%

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

      6. metadata-eval 48.8%

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

      7. metadata-eval 48.8%

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

      8. times-frac 48.8%

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

      9. *-commutative 48.8%

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

      10. times-frac 48.8%

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

      11. associate-*l/ 48.8%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in b around inf 88.1%

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

      1. fma-def 88.1%

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

      2. associate-*r/ 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*r* 88.1%

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

      5. unpow2 88.1%

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

   6. Simplified 88.1%

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

      1. fma-udef 88.1%

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

      2. associate-/l* 88.1%

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

      3. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. neg-sub0 56.1%

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

   2. associate-+l- 56.1%

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

   3. sub0-neg 56.1%

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

   4. neg-mul-1 56.1%

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

   5. associate-*r/ 56.1%

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

   6. metadata-eval 56.1%

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

   7. metadata-eval 56.1%

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

   8. times-frac 56.1%

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

   9. *-commutative 56.1%

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

   10. times-frac 56.1%

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

   11. associate-*l/ 56.1%

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

3. Simplified 56.2%

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

4. Taylor expanded in b around inf 82.2%

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

   1. fma-def 82.2%

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

   2. associate-*r/ 82.2%

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

   3. *-commutative 82.2%

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

   4. associate-*r* 82.2%

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

   5. unpow2 82.2%

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

6. Simplified 82.2%

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

   1. fma-udef 82.2%

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

   2. associate-/l* 82.2%

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

   3. *-commutative 82.2%

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

8. Applied egg-rr 82.2%

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

9. Final simplification 82.2%

   \[\leadsto -0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}} \]

Alternative 7: 64.5% 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 56.1%

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

   1. neg-sub0 56.1%

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

   2. associate-+l- 56.1%

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

   3. sub0-neg 56.1%

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

   4. neg-mul-1 56.1%

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

   5. associate-*r/ 56.1%

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

   6. metadata-eval 56.1%

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

   7. metadata-eval 56.1%

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

   8. times-frac 56.1%

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

   9. *-commutative 56.1%

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

   10. times-frac 56.1%

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

   11. associate-*l/ 56.1%

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

3. Simplified 56.2%

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

4. Taylor expanded in b around inf 64.2%

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

5. Final simplification 64.2%

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

Reproduce

herbie shell --seed 2023229 
(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)))