Quadratic roots, wide range

Percentage Accurate: 17.5% → 97.8%

Time: 8.2s

Alternatives: 5

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: 17.5% 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: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (-
   (fma
    -0.25
    (* 20.0 (/ (pow c 4.0) (/ (pow b 7.0) (pow a 3.0))))
    (/ (* (pow c 3.0) -2.0) (/ (pow b 5.0) (* a a))))
   (/ c b))
  (/ (* c (* c a)) (pow b 3.0))))

C

double code(double a, double b, double c) {
	return (fma(-0.25, (20.0 * (pow(c, 4.0) / (pow(b, 7.0) / pow(a, 3.0)))), ((pow(c, 3.0) * -2.0) / (pow(b, 5.0) / (a * a)))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(-0.25, Float64(20.0 * Float64((c ^ 4.0) / Float64((b ^ 7.0) / (a ^ 3.0)))), Float64(Float64((c ^ 3.0) * -2.0) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.25 * N[(20.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

   1. neg-sub0 18.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 18.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 18.4%

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

   4. neg-mul-1 18.4%

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

   5. associate-*l/ 18.4%

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

   6. *-commutative 18.4%

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

   7. associate-/r* 18.4%

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

   8. /-rgt-identity 18.4%

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

   9. metadata-eval 18.4%

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

3. Simplified 18.4%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in a around 0 97.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{\left({\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right) \cdot {a}^{3}}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

6. Simplified 97.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{\frac{b}{{a}^{3}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]

7. Taylor expanded in c around 0 97.7%

   \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{20 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
8. Step-by-step derivation

   1. associate-/l* 97.7%

      \[\leadsto \left(\mathsf{fma}\left(-0.25, 20 \cdot \color{blue}{\frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

9. Simplified 97.7%

   \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

10. Final simplification 97.7%

    \[\leadsto \left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

Alternative 2: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (- (/ (* (pow c 3.0) -2.0) (/ (pow b 5.0) (* a a))) (/ c b))
  (/ (* c (* c a)) (pow b 3.0))))

C

double code(double a, double b, double c) {
	return (((pow(c, 3.0) * -2.0) / (pow(b, 5.0) / (a * a))) - (c / b)) - ((c * (c * a)) / 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 ** 3.0d0) * (-2.0d0)) / ((b ** 5.0d0) / (a * a))) - (c / b)) - ((c * (c * a)) / (b ** 3.0d0))
end function

Java

public static double code(double a, double b, double c) {
	return (((Math.pow(c, 3.0) * -2.0) / (Math.pow(b, 5.0) / (a * a))) - (c / b)) - ((c * (c * a)) / Math.pow(b, 3.0));
}

Python

def code(a, b, c):
	return (((math.pow(c, 3.0) * -2.0) / (math.pow(b, 5.0) / (a * a))) - (c / b)) - ((c * (c * a)) / math.pow(b, 3.0))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((c ^ 3.0) * -2.0) / Float64((b ^ 5.0) / Float64(a * a))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((c ^ 3.0) * -2.0) / ((b ^ 5.0) / (a * a))) - (c / b)) - ((c * (c * a)) / (b ^ 3.0));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

   1. neg-sub0 18.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 18.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 18.4%

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

   4. neg-mul-1 18.4%

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

   5. associate-*l/ 18.4%

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

   6. *-commutative 18.4%

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

   7. associate-/r* 18.4%

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

   8. /-rgt-identity 18.4%

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

   9. metadata-eval 18.4%

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

3. Simplified 18.4%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 97.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
5. Step-by-step derivation

   1. +-commutative 97.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   2. mul-1-neg 97.1%

      \[\leadsto \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

   3. unsub-neg 97.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   4. +-commutative 97.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   5. mul-1-neg 97.1%

      \[\leadsto \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(-\frac{c}{b}\right)}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   6. unsub-neg 97.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} - \frac{c}{b}\right)} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   7. *-commutative 97.1%

      \[\leadsto \left(\color{blue}{\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -2} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   8. associate-/l* 97.1%

      \[\leadsto \left(\color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}} \cdot -2 - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   9. associate-*l/ 97.1%

      \[\leadsto \left(\color{blue}{\frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{{a}^{2}}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

   10. unpow2 97.1%

       \[\leadsto \left(\frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}} - \frac{c}{b}\right) - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

6. Simplified 97.1%

   \[\leadsto \color{blue}{\left(\frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]

7. Final simplification 97.1%

   \[\leadsto \left(\frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

Alternative 3: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.4%

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

   1. neg-sub0 18.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 18.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 18.4%

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

   4. neg-mul-1 18.4%

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

   5. associate-*l/ 18.4%

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

   6. *-commutative 18.4%

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

   7. associate-/r* 18.4%

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

   8. /-rgt-identity 18.4%

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

   9. metadata-eval 18.4%

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

3. Simplified 18.4%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 95.6%

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

   1. +-commutative 95.6%

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

   2. mul-1-neg 95.6%

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

   3. unsub-neg 95.6%

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

   4. associate-*r/ 95.6%

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

   5. neg-mul-1 95.6%

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

   6. unpow2 95.6%

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

   7. associate-*l* 95.6%

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

6. Simplified 95.6%

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

7. Final simplification 95.6%

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

Alternative 4: 95.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(4 \cdot a\right)}{\left(b + b\right) + -2 \cdot \left(a \cdot \frac{c}{b}\right)} \cdot \frac{-0.5}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ (* c (* 4.0 a)) (+ (+ b b) (* -2.0 (* a (/ c b))))) (/ -0.5 a)))

C

double code(double a, double b, double c) {
	return ((c * (4.0 * a)) / ((b + b) + (-2.0 * (a * (c / b))))) * (-0.5 / 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 = ((c * (4.0d0 * a)) / ((b + b) + ((-2.0d0) * (a * (c / b))))) * ((-0.5d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (4.0 * a)) / ((b + b) + (-2.0 * (a * (c / b))))) * (-0.5 / a);
}

Python

def code(a, b, c):
	return ((c * (4.0 * a)) / ((b + b) + (-2.0 * (a * (c / b))))) * (-0.5 / a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(4.0 * a)) / Float64(Float64(b + b) + Float64(-2.0 * Float64(a * Float64(c / b))))) * Float64(-0.5 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (4.0 * a)) / ((b + b) + (-2.0 * (a * (c / b))))) * (-0.5 / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b + b), $MachinePrecision] + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

   1. neg-sub0 18.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 18.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 18.4%

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

   4. neg-mul-1 18.4%

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

   5. associate-*l/ 18.4%

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

   6. *-commutative 18.4%

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

   7. associate-/r* 18.4%

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

   8. /-rgt-identity 18.4%

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

   9. metadata-eval 18.4%

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

3. Simplified 18.4%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in a around 0 13.8%

   \[\leadsto \left(b - \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}\right) \cdot \frac{-0.5}{a} \]
5. Step-by-step derivation

   1. *-commutative 13.8%

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

   2. associate-/l* 13.8%

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

6. Simplified 13.8%

   \[\leadsto \left(b - \color{blue}{\left(b + \frac{c}{\frac{b}{a}} \cdot -2\right)}\right) \cdot \frac{-0.5}{a} \]
7. Step-by-step derivation

   1. flip-- 13.7%

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

   2. associate-/l* 13.7%

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

   3. *-commutative 13.7%

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

   4. associate-/l* 13.7%

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

   5. *-commutative 13.7%

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

   6. associate-/l* 13.7%

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

   7. *-commutative 13.7%

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

8. Applied egg-rr 13.7%

   \[\leadsto \color{blue}{\frac{b \cdot b - \left(b + -2 \cdot \frac{c \cdot a}{b}\right) \cdot \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}{b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}} \cdot \frac{-0.5}{a} \]
9. Step-by-step derivation

   1. difference-of-squares 13.8%

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

   2. associate-+r+ 13.8%

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

   3. associate-*l/ 13.8%

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

   4. *-commutative 13.8%

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

   5. associate--r+ 89.4%

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

   6. associate-*l/ 89.5%

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

   7. *-commutative 89.5%

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

   8. associate-+r+ 89.5%

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

   9. associate-*l/ 89.5%

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

   10. *-commutative 89.5%

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

10. Simplified 89.5%

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

11. Taylor expanded in b around inf 95.2%

    \[\leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{\left(b + b\right) + -2 \cdot \left(a \cdot \frac{c}{b}\right)} \cdot \frac{-0.5}{a} \]
12. Step-by-step derivation

    1. *-commutative 95.2%

       \[\leadsto \frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{\left(b + b\right) + -2 \cdot \left(a \cdot \frac{c}{b}\right)} \cdot \frac{-0.5}{a} \]

    2. associate-*l* 95.2%

       \[\leadsto \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(b + b\right) + -2 \cdot \left(a \cdot \frac{c}{b}\right)} \cdot \frac{-0.5}{a} \]

13. Simplified 95.2%

    \[\leadsto \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(b + b\right) + -2 \cdot \left(a \cdot \frac{c}{b}\right)} \cdot \frac{-0.5}{a} \]

14. Final simplification 95.2%

    \[\leadsto \frac{c \cdot \left(4 \cdot a\right)}{\left(b + b\right) + -2 \cdot \left(a \cdot \frac{c}{b}\right)} \cdot \frac{-0.5}{a} \]

Alternative 5: 90.7% 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(Float64(-c) / 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 18.4%

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

   1. neg-sub0 18.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 18.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 18.4%

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

   4. neg-mul-1 18.4%

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

   5. associate-*l/ 18.4%

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

   6. *-commutative 18.4%

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

   7. associate-/r* 18.4%

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

   8. /-rgt-identity 18.4%

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

   9. metadata-eval 18.4%

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

3. Simplified 18.4%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 89.9%

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

   1. associate-*r/ 89.9%

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

   2. neg-mul-1 89.9%

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

6. Simplified 89.9%

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

7. Final simplification 89.9%

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

Reproduce

herbie shell --seed 2023214 
(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)))