Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 12.7s

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: 18.0% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

   1. neg-sub0 15.0%

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

   2. associate-+l- 15.0%

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

   3. sub0-neg 15.0%

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

   4. neg-mul-1 15.0%

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

   5. associate-*l/ 15.0%

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

   6. *-commutative 15.0%

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

   7. associate-/r* 15.0%

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

   8. /-rgt-identity 15.0%

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

   9. metadata-eval 15.0%

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

3. Simplified 15.0%

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

   1. fma-udef 15.0%

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

   2. associate-*l* 15.0%

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

   3. *-commutative 15.0%

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

5. Applied egg-rr 15.0%

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

   1. add-cbrt-cube 15.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}} \cdot \frac{-0.5}{a} \]

   2. fma-def 15.0%

      \[\leadsto \sqrt[3]{\left(\left(b - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)} \cdot \frac{-0.5}{a} \]

   3. *-commutative 15.0%

      \[\leadsto \sqrt[3]{\left(\left(b - \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, b \cdot b\right)}\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)} \cdot \frac{-0.5}{a} \]

   4. fma-def 15.0%

      \[\leadsto \sqrt[3]{\left(\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)} \cdot \frac{-0.5}{a} \]

   5. *-commutative 15.0%

      \[\leadsto \sqrt[3]{\left(\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, b \cdot b\right)}\right)\right) \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)} \cdot \frac{-0.5}{a} \]

   6. fma-def 15.0%

      \[\leadsto \sqrt[3]{\left(\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)\right) \cdot \left(b - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)} \cdot \frac{-0.5}{a} \]

   7. *-commutative 15.0%

      \[\leadsto \sqrt[3]{\left(\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, b \cdot b\right)}\right)} \cdot \frac{-0.5}{a} \]

7. Applied egg-rr 15.0%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}} \cdot \frac{-0.5}{a} \]
8. Step-by-step derivation

   1. associate-*l* 15.0%

      \[\leadsto \sqrt[3]{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)\right)}} \cdot \frac{-0.5}{a} \]

   2. cube-unmult 15.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}^{3}}} \cdot \frac{-0.5}{a} \]

9. Simplified 15.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}^{3}}} \cdot \frac{-0.5}{a} \]

10. Taylor expanded in b around inf 98.5%

    \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 2: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

   1. neg-sub0 15.0%

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

   2. associate-+l- 15.0%

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

   3. sub0-neg 15.0%

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

   4. neg-mul-1 15.0%

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

   5. associate-*l/ 15.0%

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

   6. *-commutative 15.0%

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

   7. associate-/r* 15.0%

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

   8. /-rgt-identity 15.0%

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

   9. metadata-eval 15.0%

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

3. Simplified 15.0%

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

   1. fma-udef 15.0%

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

   2. associate-*l* 15.0%

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

   3. *-commutative 15.0%

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

5. Applied egg-rr 15.0%

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

6. Taylor expanded in b around inf 97.7%

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

   1. +-commutative 97.7%

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

   2. mul-1-neg 97.7%

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

   3. unpow2 97.7%

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

   4. unsub-neg 97.7%

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

   5. +-commutative 97.7%

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

   6. mul-1-neg 97.7%

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

   7. unsub-neg 97.7%

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

   8. associate-/l* 97.7%

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

   9. associate-/r/ 97.7%

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

   10. unpow2 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 3: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

   1. neg-sub0 15.0%

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

   2. associate-+l- 15.0%

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

   3. sub0-neg 15.0%

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

   4. neg-mul-1 15.0%

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

   5. associate-*l/ 15.0%

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

   6. *-commutative 15.0%

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

   7. associate-/r* 15.0%

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

   8. /-rgt-identity 15.0%

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

   9. metadata-eval 15.0%

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

3. Simplified 15.0%

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

4. Taylor expanded in b around inf 96.4%

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

   1. distribute-lft-out 96.4%

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

   2. mul-1-neg 96.4%

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

   3. +-commutative 96.4%

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

   4. unpow2 96.4%

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

   5. associate-*l* 96.4%

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

6. Simplified 96.4%

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

7. Final simplification 96.4%

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

Alternative 4: 90.3% 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 15.0%

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

   1. neg-sub0 15.0%

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

   2. associate-+l- 15.0%

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

   3. sub0-neg 15.0%

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

   4. neg-mul-1 15.0%

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

   5. associate-*l/ 15.0%

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

   6. *-commutative 15.0%

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

   7. associate-/r* 15.0%

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

   8. /-rgt-identity 15.0%

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

   9. metadata-eval 15.0%

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

3. Simplified 15.0%

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

4. Taylor expanded in b around inf 92.4%

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

   1. associate-*r/ 92.4%

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

   2. neg-mul-1 92.4%

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

6. Simplified 92.4%

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

7. Final simplification 92.4%

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

Alternative 5: 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 15.0%

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

   1. add-log-exp 8.9%

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

   2. neg-mul-1 8.9%

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

   3. fma-def 8.9%

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

   4. *-commutative 8.9%

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

   5. *-commutative 8.9%

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

3. Applied egg-rr 8.9%

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

4. Taylor expanded in c around 0 3.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. 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%

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

6. Simplified 3.3%

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

7. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2023240 
(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)))