Cubic critical, wide range

Percentage Accurate: 17.3% → 97.7%

Time: 19.0s

Alternatives: 10

Speedup: 23.2×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (*
     a
     (+
      (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
      (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
}

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 / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0)))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0)))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0)))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 7.0)))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 98.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

4. Taylor expanded in c around 0 98.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]

5. Final simplification 98.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
6. Add Preprocessing

Alternative 2: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
}

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 / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0)))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 98.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]

4. Final simplification 98.2%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right) \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
     (* -0.375 (/ a (pow b 3.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * (((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * (a / (b ** 3.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + (-0.375 * (a / Math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) + (-0.375 * (a / math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.5625 * ((c * (a ^ 2.0)) / (b ^ 5.0))) + (-0.375 * (a / (b ^ 3.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 98.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Final simplification 98.0%

   \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (* -0.5 c) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b))

C

double code(double a, double b, double c) {
	return ((-0.5 * c) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-0.5d0) * c) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0)))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-0.5 * c) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 2.0)))) / b;
}

Python

def code(a, b, c):
	return ((-0.5 * c) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 2.0)))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-0.5 * c) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-0.5 * c) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 2.0)))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5 * c), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 96.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

4. Final simplification 96.7%

   \[\leadsto \frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{{b}^{2}}, -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ c b) (fma a (* -0.375 (/ c (pow b 2.0))) -0.5)))

C

double code(double a, double b, double c) {
	return (c / b) * fma(a, (-0.375 * (c / pow(b, 2.0))), -0.5);
}

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * fma(a, Float64(-0.375 * Float64(c / (b ^ 2.0))), -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * N[(a * N[(-0.375 * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in b around inf 96.4%

   \[\leadsto c \cdot \color{blue}{\frac{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}{b}} \]
5. Step-by-step derivation

   1. associate-*r/ 96.7%

      \[\leadsto \color{blue}{\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}} \]

   2. *-commutative 96.7%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{2}} \cdot -0.375} - 0.5\right)}{b} \]

   3. fmm-def 96.7%

      \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{{b}^{2}}, -0.375, -0.5\right)}}{b} \]

   4. associate-/l* 96.7%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, -0.375, -0.5\right)}{b} \]

   5. metadata-eval 96.7%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, \color{blue}{-0.5}\right)}{b} \]

6. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\frac{c \cdot \mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, -0.5\right)}{b}} \]
7. Step-by-step derivation

   1. *-commutative 96.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, -0.5\right) \cdot c}}{b} \]

   2. associate-/l* 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, -0.5\right) \cdot \frac{c}{b}} \]

   3. fma-undefine 96.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot -0.375 + -0.5\right)} \cdot \frac{c}{b} \]

   4. associate-*l* 96.7%

      \[\leadsto \left(\color{blue}{a \cdot \left(\frac{c}{{b}^{2}} \cdot -0.375\right)} + -0.5\right) \cdot \frac{c}{b} \]

   5. fma-define 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{c}{{b}^{2}} \cdot -0.375, -0.5\right)} \cdot \frac{c}{b} \]

   6. *-commutative 96.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{-0.375 \cdot \frac{c}{{b}^{2}}}, -0.5\right) \cdot \frac{c}{b} \]

8. Simplified 96.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, -0.375 \cdot \frac{c}{{b}^{2}}, -0.5\right) \cdot \frac{c}{b}} \]

9. Final simplification 96.7%

   \[\leadsto \frac{c}{b} \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{{b}^{2}}, -0.5\right) \]
10. Add Preprocessing

Alternative 6: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma (* a (/ c (pow b 2.0))) -0.375 -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma((a * (c / pow(b, 2.0))), -0.375, -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(Float64(a * Float64(c / (b ^ 2.0))), -0.375, -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(a * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in b around inf 96.4%

   \[\leadsto c \cdot \color{blue}{\frac{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}{b}} \]
5. Step-by-step derivation

   1. associate-*r/ 96.7%

      \[\leadsto \color{blue}{\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}} \]

   2. *-commutative 96.7%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{2}} \cdot -0.375} - 0.5\right)}{b} \]

   3. fmm-def 96.7%

      \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(\frac{a \cdot c}{{b}^{2}}, -0.375, -0.5\right)}}{b} \]

   4. associate-/l* 96.7%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{a \cdot \frac{c}{{b}^{2}}}, -0.375, -0.5\right)}{b} \]

   5. metadata-eval 96.7%

      \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, \color{blue}{-0.5}\right)}{b} \]

6. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\frac{c \cdot \mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, -0.5\right)}{b}} \]

7. Final simplification 96.7%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(a \cdot \frac{c}{{b}^{2}}, -0.375, -0.5\right)}{b} \]
8. Add Preprocessing

Alternative 7: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (- (* -0.375 (/ (* c a) (pow b 2.0))) 0.5) b)))

C

double code(double a, double b, double c) {
	return c * (((-0.375 * ((c * a) / pow(b, 2.0))) - 0.5) / 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 * ((((-0.375d0) * ((c * a) / (b ** 2.0d0))) - 0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (((-0.375 * ((c * a) / Math.pow(b, 2.0))) - 0.5) / b);
}

Python

def code(a, b, c):
	return c * (((-0.375 * ((c * a) / math.pow(b, 2.0))) - 0.5) / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 2.0))) - 0.5) / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((-0.375 * ((c * a) / (b ^ 2.0))) - 0.5) / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in b around inf 96.4%

   \[\leadsto c \cdot \color{blue}{\frac{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}{b}} \]

5. Final simplification 96.4%

   \[\leadsto c \cdot \frac{-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5}{b} \]
6. Add Preprocessing

Alternative 8: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\left(c \cdot a\right) \cdot \left(-0.375 \cdot {b}^{-3}\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* (* c a) (* -0.375 (pow b -3.0))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * (((c * a) * (-0.375 * pow(b, -3.0))) - (0.5 / 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 * (((c * a) * ((-0.375d0) * (b ** (-3.0d0)))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * (((c * a) * (-0.375 * Math.pow(b, -3.0))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * (((c * a) * (-0.375 * math.pow(b, -3.0))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(c * a) * Float64(-0.375 * (b ^ -3.0))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((c * a) * (-0.375 * (b ^ -3.0))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(c * a), $MachinePrecision] * N[(-0.375 * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. log1p-expm1-u 82.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)\right)\right)} \]

   2. log1p-undefine 20.7%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)\right)\right)} \]

   3. *-commutative 20.7%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{3}} \cdot -0.375} - 0.5 \cdot \frac{1}{b}\right)\right)\right) \]

   4. div-inv 20.7%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(c \cdot \left(\color{blue}{\left(\left(a \cdot c\right) \cdot \frac{1}{{b}^{3}}\right)} \cdot -0.375 - 0.5 \cdot \frac{1}{b}\right)\right)\right) \]

   5. pow-flip 20.7%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(c \cdot \left(\left(\left(a \cdot c\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) \cdot -0.375 - 0.5 \cdot \frac{1}{b}\right)\right)\right) \]

   6. metadata-eval 20.7%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(c \cdot \left(\left(\left(a \cdot c\right) \cdot {b}^{\color{blue}{-3}}\right) \cdot -0.375 - 0.5 \cdot \frac{1}{b}\right)\right)\right) \]

   7. un-div-inv 20.7%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(c \cdot \left(\left(\left(a \cdot c\right) \cdot {b}^{-3}\right) \cdot -0.375 - \color{blue}{\frac{0.5}{b}}\right)\right)\right) \]

5. Applied egg-rr 20.7%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(c \cdot \left(\left(\left(a \cdot c\right) \cdot {b}^{-3}\right) \cdot -0.375 - \frac{0.5}{b}\right)\right)\right)} \]
6. Step-by-step derivation

   1. log1p-define 82.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(c \cdot \left(\left(\left(a \cdot c\right) \cdot {b}^{-3}\right) \cdot -0.375 - \frac{0.5}{b}\right)\right)\right)} \]

   2. log1p-expm1-u 96.4%

      \[\leadsto \color{blue}{c \cdot \left(\left(\left(a \cdot c\right) \cdot {b}^{-3}\right) \cdot -0.375 - \frac{0.5}{b}\right)} \]

   3. *-commutative 96.4%

      \[\leadsto \color{blue}{\left(\left(\left(a \cdot c\right) \cdot {b}^{-3}\right) \cdot -0.375 - \frac{0.5}{b}\right) \cdot c} \]

   4. associate-*l* 96.4%

      \[\leadsto \left(\color{blue}{\left(a \cdot c\right) \cdot \left({b}^{-3} \cdot -0.375\right)} - \frac{0.5}{b}\right) \cdot c \]

7. Applied egg-rr 96.4%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \left({b}^{-3} \cdot -0.375\right) - \frac{0.5}{b}\right) \cdot c} \]

8. Final simplification 96.4%

   \[\leadsto c \cdot \left(\left(c \cdot a\right) \cdot \left(-0.375 \cdot {b}^{-3}\right) - \frac{0.5}{b}\right) \]
9. Add Preprocessing

Alternative 9: 90.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 96.4%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in c around 0 91.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
5. Step-by-step derivation

   1. associate-*r/ 91.1%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. *-commutative 91.1%

      \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

   3. associate-/l* 90.8%

      \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b}} \]

6. Simplified 90.8%

   \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b}} \]

7. Final simplification 90.8%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
8. Add Preprocessing

Alternative 10: 90.8% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 91.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 91.1%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. *-commutative 91.1%

      \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

5. Simplified 91.1%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

6. Final simplification 91.1%

   \[\leadsto \frac{-0.5 \cdot c}{b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))