Quadratic roots, wide range

Percentage Accurate: 18.1% → 99.7%

Time: 13.7s

Alternatives: 6

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 18.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{a \cdot c}{a} \cdot 2}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (/ (* a c) a) 2.0)
  (- (- b) (sqrt (* c (fma a -4.0 (/ (pow b 2.0) c)))))))

C

double code(double a, double b, double c) {
	return (((a * c) / a) * 2.0) / (-b - sqrt((c * fma(a, -4.0, (pow(b, 2.0) / c)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(a * c) / a) * 2.0) / Float64(Float64(-b) - sqrt(Float64(c * fma(a, -4.0, Float64((b ^ 2.0) / c))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * c), $MachinePrecision] / a), $MachinePrecision] * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0 + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 15.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 15.2%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around inf 15.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. flip-+ 15.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)} \cdot \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}}{a \cdot 2} \]

   2. pow2 15.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)} \cdot \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   3. add-sqr-sqrt 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   4. cancel-sign-sub-inv 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(-4\right) \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   5. metadata-eval 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-4} \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   6. cancel-sign-sub-inv 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(-4\right) \cdot a\right)}}}}{a \cdot 2} \]

   7. metadata-eval 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-4} \cdot a\right)}}}{a \cdot 2} \]

7. Applied egg-rr 15.8%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}}{a \cdot 2} \]

8. Taylor expanded in b around 0 99.3%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]

   2. *-commutative 99.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right)} \cdot 4}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]

10. Simplified 99.3%

    \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. *-un-lft-identity 99.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2}} \]

    2. associate-/l/ 99.4%

       \[\leadsto 1 \cdot \color{blue}{\frac{\left(c \cdot a\right) \cdot 4}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}\right)}} \]

    3. associate-*l* 99.4%

       \[\leadsto 1 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}\right)} \]

    4. +-commutative 99.4%

       \[\leadsto 1 \cdot \frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \color{blue}{\left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}}\right)} \]

    5. *-commutative 99.4%

       \[\leadsto 1 \cdot \frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(\color{blue}{a \cdot -4} + \frac{{b}^{2}}{c}\right)}\right)} \]

    6. fma-define 99.4%

       \[\leadsto 1 \cdot \frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \color{blue}{\mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}}\right)} \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{1 \cdot \frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}\right)}} \]
13. Step-by-step derivation

    1. *-lft-identity 99.4%

       \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}\right)}} \]

    2. associate-/r* 99.7%

       \[\leadsto \color{blue}{\frac{\frac{c \cdot \left(a \cdot 4\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}}} \]

    3. associate-*r* 99.7%

       \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{a \cdot 2}}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}} \]

    4. times-frac 99.7%

       \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{a} \cdot \frac{4}{2}}}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}} \]

    5. *-commutative 99.7%

       \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{a} \cdot \frac{4}{2}}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}} \]

    6. metadata-eval 99.7%

       \[\leadsto \frac{\frac{a \cdot c}{a} \cdot \color{blue}{2}}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}} \]

14. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\frac{a \cdot c}{a} \cdot 2}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}}} \]

15. Final simplification 99.7%

    \[\leadsto \frac{\frac{a \cdot c}{a} \cdot 2}{\left(-b\right) - \sqrt{c \cdot \mathsf{fma}\left(a, -4, \frac{{b}^{2}}{c}\right)}} \]
16. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{{b}^{2} + \left(a \cdot c\right) \cdot -4}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* (* a c) 4.0) (- (- b) (sqrt (+ (pow b 2.0) (* (* a c) -4.0)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (((a * c) * 4.0) / (-b - sqrt((pow(b, 2.0) + ((a * c) * -4.0))))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((a * c) * 4.0d0) / (-b - sqrt(((b ** 2.0d0) + ((a * c) * (-4.0d0)))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (((a * c) * 4.0) / (-b - Math.sqrt((Math.pow(b, 2.0) + ((a * c) * -4.0))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (((a * c) * 4.0) / (-b - math.sqrt((math.pow(b, 2.0) + ((a * c) * -4.0))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(a * c) * 4.0) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) + Float64(Float64(a * c) * -4.0))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((a * c) * 4.0) / (-b - sqrt(((b ^ 2.0) + ((a * c) * -4.0))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 15.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 15.2%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around inf 15.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. flip-+ 15.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)} \cdot \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}}{a \cdot 2} \]

   2. pow2 15.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)} \cdot \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   3. add-sqr-sqrt 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   4. cancel-sign-sub-inv 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(-4\right) \cdot a\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   5. metadata-eval 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-4} \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]

   6. cancel-sign-sub-inv 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(-4\right) \cdot a\right)}}}}{a \cdot 2} \]

   7. metadata-eval 15.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-4} \cdot a\right)}}}{a \cdot 2} \]

7. Applied egg-rr 15.8%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}}{a \cdot 2} \]

8. Taylor expanded in b around 0 99.3%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]

   2. *-commutative 99.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right)} \cdot 4}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]

10. Simplified 99.3%

    \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(-b\right) - \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}}}{a \cdot 2} \]

11. Taylor expanded in c around 0 99.3%

    \[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}}}{a \cdot 2} \]

12. Final simplification 99.3%

    \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{{b}^{2} + \left(a \cdot c\right) \cdot -4}}}{a \cdot 2} \]
13. Add Preprocessing

Alternative 3: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 15.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 15.2%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 96.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
6. Step-by-step derivation

   1. mul-1-neg 96.8%

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

   2. unsub-neg 96.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   3. associate-*r/ 96.8%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   4. mul-1-neg 96.8%

      \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   5. associate-/l* 96.8%

      \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

7. Simplified 96.8%

   \[\leadsto \color{blue}{\frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

8. Final simplification 96.8%

   \[\leadsto \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \]
9. Add Preprocessing

Alternative 4: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 15.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 15.2%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 96.5%

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

   1. associate-*r/ 96.5%

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

   2. neg-mul-1 96.5%

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

   3. distribute-rgt-neg-in 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in c around inf 96.2%

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

   1. mul-1-neg 96.2%

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

   2. *-commutative 96.2%

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

10. Simplified 96.2%

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

11. Taylor expanded in b around inf 96.8%

    \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
12. Step-by-step derivation

    1. distribute-lft-out 96.8%

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

    2. associate-*r/ 96.8%

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

    3. mul-1-neg 96.8%

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

    4. distribute-neg-frac2 96.8%

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

    5. associate-/l* 96.8%

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

    6. unpow2 96.8%

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

    7. unpow2 96.8%

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

    8. times-frac 96.8%

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

    9. unpow2 96.8%

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

13. Simplified 96.8%

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

14. Final simplification 96.8%

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

Alternative 5: 90.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 15.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 15.2%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 92.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 92.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 92.4%

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

7. Simplified 92.4%

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

8. Final simplification 92.4%

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

Alternative 6: 3.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 15.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 15.2%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around inf 15.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. div-inv 15.1%

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}} \]

   2. neg-mul-1 15.1%

      \[\leadsto \left(\color{blue}{-1 \cdot b} + \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2} \]

   3. fma-define 15.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(\frac{{b}^{2}}{c} - 4 \cdot a\right)}\right)} \cdot \frac{1}{a \cdot 2} \]

   4. cancel-sign-sub-inv 15.1%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{c \cdot \color{blue}{\left(\frac{{b}^{2}}{c} + \left(-4\right) \cdot a\right)}}\right) \cdot \frac{1}{a \cdot 2} \]

   5. metadata-eval 15.1%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + \color{blue}{-4} \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2} \]

7. Applied egg-rr 15.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(\frac{{b}^{2}}{c} + -4 \cdot a\right)}\right) \cdot \frac{1}{a \cdot 2}} \]

8. Taylor expanded in c around 0 3.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.3%

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

   2. distribute-rgt1-in 3.3%

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

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.3%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2024080 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))