Quadratic roots, medium range

Percentage Accurate: 31.4% → 95.5%

Time: 13.2s

Alternatives: 4

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

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: 31.4% 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: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

2. Taylor expanded in a around 0 96.2%

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

4. Simplified 96.2%

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

5. Taylor expanded in c around 0 96.2%

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

   1. associate-/l* 96.2%

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

   2. associate-*r/ 96.2%

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

7. Simplified 96.2%

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

8. Final simplification 96.2%

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

Alternative 2: 93.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

2. Taylor expanded in b around inf 94.8%

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

   1. associate-+r+ 94.8%

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

   2. mul-1-neg 94.8%

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

   3. unsub-neg 94.8%

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

   4. mul-1-neg 94.8%

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

   5. unsub-neg 94.8%

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

   6. associate-/l* 94.8%

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

   7. associate-*r/ 94.8%

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

   8. unpow2 94.8%

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

   9. associate-/l* 94.8%

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

   10. associate-/r/ 94.8%

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

   11. unpow2 94.8%

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

4. Simplified 94.8%

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

5. Final simplification 94.8%

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

Alternative 3: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

2. Taylor expanded in b around inf 91.5%

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

   1. mul-1-neg 91.5%

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

   2. unsub-neg 91.5%

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

   3. mul-1-neg 91.5%

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

   4. distribute-neg-frac 91.5%

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

   5. associate-/l* 91.5%

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

   6. associate-/r/ 91.5%

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

   7. unpow2 91.5%

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

4. Simplified 91.5%

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

5. Final simplification 91.5%

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

Alternative 4: 81.3% 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 31.9%

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

2. Taylor expanded in b around inf 80.8%

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

   1. mul-1-neg 80.8%

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

   2. distribute-neg-frac 80.8%

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

4. Simplified 80.8%

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

5. Final simplification 80.8%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))