Quadratic roots, wide range

Percentage Accurate: 18.1% → 97.7%

Time: 12.2s

Alternatives: 10

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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

4. Taylor expanded in a around 0 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in b around 0 98.0%

   \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
8. Step-by-step derivation

   1. associate-/l* 98.0%

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

   2. distribute-rgt-out 98.0%

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

   3. metadata-eval 98.0%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 2: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

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. fma-def 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-/l* 97.1%

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

   8. unpow2 97.1%

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

   9. associate-*r/ 97.1%

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

   10. neg-mul-1 97.1%

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

   11. unpow2 97.1%

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

   12. associate-*l* 97.1%

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

6. Simplified 97.1%

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

7. Final simplification 97.1%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (pow
  (+ (/ a b) (- (* -2.0 (/ (* (* c (* a a)) -0.5) (pow b 3.0))) (/ b c)))
  -1.0))

C

double code(double a, double b, double c) {
	return pow(((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / pow(b, 3.0))) - (b / c))), -1.0);
}

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 / b) + (((-2.0d0) * (((c * (a * a)) * (-0.5d0)) / (b ** 3.0d0))) - (b / c))) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / Math.pow(b, 3.0))) - (b / c))), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / math.pow(b, 3.0))) - (b / c))), -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(a / b) + Float64(Float64(-2.0 * Float64(Float64(Float64(c * Float64(a * a)) * -0.5) / (b ^ 3.0))) - Float64(b / c))) ^ -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a / b) + ((-2.0 * (((c * (a * a)) * -0.5) / (b ^ 3.0))) - (b / c))) ^ -1.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a / b), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 17.9%

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

   1. clear-num 17.9%

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

   2. inv-pow 17.9%

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

   3. *-commutative 17.9%

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

   4. neg-mul-1 17.9%

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

   5. fma-def 17.9%

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

   6. *-commutative 17.9%

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

   7. *-commutative 17.9%

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

3. Applied egg-rr 17.9%

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

4. Taylor expanded in b around inf 96.8%

   \[\leadsto {\color{blue}{\left(\frac{a}{b} + \left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + -1 \cdot \frac{b}{c}\right)\right)}}^{-1} \]
5. Step-by-step derivation

   1. mul-1-neg 96.8%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \color{blue}{\left(-\frac{b}{c}\right)}\right)\right)}^{-1} \]

   2. unsub-neg 96.8%

      \[\leadsto {\left(\frac{a}{b} + \color{blue}{\left(-2 \cdot \frac{0.5 \cdot \left(c \cdot {a}^{2}\right) + -1 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} - \frac{b}{c}\right)}\right)}^{-1} \]

   3. distribute-rgt-out 96.8%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\color{blue}{\left(c \cdot {a}^{2}\right) \cdot \left(0.5 + -1\right)}}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

   4. unpow2 96.8%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(0.5 + -1\right)}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

   5. metadata-eval 96.8%

      \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-0.5}}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

6. Simplified 96.8%

   \[\leadsto {\color{blue}{\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}} - \frac{b}{c}\right)\right)}}^{-1} \]

7. Final simplification 96.8%

   \[\leadsto {\left(\frac{a}{b} + \left(-2 \cdot \frac{\left(c \cdot \left(a \cdot a\right)\right) \cdot -0.5}{{b}^{3}} - \frac{b}{c}\right)\right)}^{-1} \]

Alternative 4: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

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%

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

Alternative 5: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (pow (- (/ a b) (/ b c)) -1.0))

C

double code(double a, double b, double c) {
	return pow(((a / b) - (b / c)), -1.0);
}

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 / b) - (b / c)) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((a / b) - (b / c)), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((a / b) - (b / c)), -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(a / b) - Float64(b / c)) ^ -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a / b) - (b / c)) ^ -1.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 17.9%

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

   1. clear-num 17.9%

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

   2. inv-pow 17.9%

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

   3. *-commutative 17.9%

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

   4. neg-mul-1 17.9%

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

   5. fma-def 17.9%

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

   6. *-commutative 17.9%

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

   7. *-commutative 17.9%

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

3. Applied egg-rr 17.9%

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

4. Taylor expanded in b around inf 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

6. Simplified 95.3%

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

7. Final simplification 95.3%

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

Alternative 6: 94.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

4. Taylor expanded in a around 0 13.0%

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

   1. *-commutative 13.0%

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

   2. associate-/l* 13.0%

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

6. Simplified 13.0%

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

   1. flip-- 12.9%

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

   2. associate-/r/ 12.9%

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

   3. associate-/r/ 12.9%

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

   4. associate-/r/ 12.9%

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

8. Applied egg-rr 12.9%

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

   1. associate-*l* 12.9%

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

   2. associate-*l* 12.9%

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

   3. associate-*l* 12.9%

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

10. Simplified 12.9%

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

11. Taylor expanded in b around inf 95.2%

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

    1. *-commutative 95.2%

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

    2. associate-*r* 95.2%

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

    3. *-commutative 95.2%

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

13. Simplified 95.2%

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

14. Final simplification 95.2%

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

Alternative 7: 95.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

4. Taylor expanded in a around 0 13.0%

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

   1. *-commutative 13.0%

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

   2. associate-/l* 13.0%

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

6. Simplified 13.0%

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

   1. flip-- 12.9%

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

   2. associate-/r/ 12.9%

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

   3. associate-/r/ 12.9%

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

   4. associate-/r/ 12.9%

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

8. Applied egg-rr 12.9%

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

   1. associate-*l* 12.9%

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

   2. associate-*l* 12.9%

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

   3. associate-*l* 12.9%

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

10. Simplified 12.9%

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

11. Taylor expanded in b around inf 95.2%

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

    1. *-commutative 95.2%

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

    2. associate-*r* 95.2%

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

    3. *-commutative 95.2%

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

13. Simplified 95.2%

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

    1. frac-times 95.3%

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

    2. *-commutative 95.3%

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

15. Applied egg-rr 95.3%

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

16. Final simplification 95.3%

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

Alternative 8: 95.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

4. Taylor expanded in a around 0 13.0%

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

   1. *-commutative 13.0%

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

   2. associate-/l* 13.0%

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

6. Simplified 13.0%

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

   1. flip-- 12.9%

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

   2. associate-/r/ 12.9%

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

   3. associate-/r/ 12.9%

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

   4. associate-/r/ 12.9%

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

8. Applied egg-rr 12.9%

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

   1. associate-*l* 12.9%

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

   2. associate-*l* 12.9%

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

   3. associate-*l* 12.9%

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

10. Simplified 12.9%

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

11. Taylor expanded in b around inf 95.2%

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

    1. *-commutative 95.2%

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

    2. associate-*r* 95.2%

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

    3. *-commutative 95.2%

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

13. Simplified 95.2%

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

    1. associate-*l/ 95.3%

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

    2. *-commutative 95.3%

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

15. Applied egg-rr 95.3%

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

16. Final simplification 95.3%

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

Alternative 9: 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 17.9%

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

   1. neg-sub0 17.9%

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

   2. associate-+l- 17.9%

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

   3. sub0-neg 17.9%

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

   4. neg-mul-1 17.9%

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

   5. associate-*l/ 17.9%

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

   6. *-commutative 17.9%

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

   7. associate-/r* 17.9%

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

   8. /-rgt-identity 17.9%

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

   9. metadata-eval 17.9%

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

3. Simplified 17.9%

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

4. Taylor expanded in b around inf 90.6%

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

   1. associate-*r/ 90.6%

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

   2. neg-mul-1 90.6%

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

6. Simplified 90.6%

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

7. Final simplification 90.6%

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

Alternative 10: 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 17.9%

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

   1. add-cube-cbrt 17.9%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

   2. pow3 17.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{3}}}{2 \cdot a} \]

   3. neg-mul-1 17.9%

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

   4. fma-def 17.9%

      \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{3}}{2 \cdot a} \]

   5. *-commutative 17.9%

      \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{3}}{2 \cdot a} \]

   6. *-commutative 17.9%

      \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{3}}{2 \cdot a} \]

3. Applied egg-rr 17.9%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

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. neg-mul-1 3.3%

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

   3. sub-neg 3.3%

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

   4. +-inverses 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 2023203 
(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)))