Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Percentage Accurate: 69.4% → 76.1%

Time: 15.1s

Alternatives: 8

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a}{b \cdot -3}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* 2.0 (sqrt x)) (cos y) (/ a (* b -3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((2.0 * sqrt(x)), cos(y), (a / (b * -3.0)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(2.0 * sqrt(x)), cos(y), Float64(a / Float64(b * -3.0)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 73.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Applied rewrites 73.6%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3}} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{a}{b \cdot 3}}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{a}{\color{blue}{b \cdot 3}}\right)\right) \]

   10. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{\frac{a}{b}}{3}}\right)\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{\color{blue}{\frac{a}{b}}}{3}\right)\right) \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

   14. metadata-eval 73.6

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{\color{blue}{-3}}\right) \]

7. Applied rewrites 73.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{-3}\right)} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{-3}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\color{blue}{\frac{a}{b}}}{-3}\right) \]

   3. associate-/l/ N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{-3 \cdot b}}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot b}\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\mathsf{neg}\left(3 \cdot b\right)}}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{\mathsf{neg}\left(b \cdot 3\right)}}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{3 \cdot b}\right)}\right) \]

   11. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot b}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{-3} \cdot b}\right) \]

   13. lower-*.f64 73.6

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{-3 \cdot b}}\right) \]

9. Applied rewrites 73.6%

   \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{-3 \cdot b}}\right) \]

10. Final simplification 73.6%

    \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \frac{a}{b \cdot -3}\right) \]
11. Add Preprocessing

Alternative 2: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := \mathsf{fma}\left(t\_2, 1, \frac{a}{b \cdot -3}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-22}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (* 2.0 (sqrt x)))
        (t_3 (fma t_2 1.0 (/ a (* b -3.0)))))
   (if (<= t_1 -5e-22)
     t_3
     (if (<= t_1 5e-74)
       (* (cos (fma -0.3333333333333333 (* t z) y)) t_2)
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = 2.0 * sqrt(x);
	double t_3 = fma(t_2, 1.0, (a / (b * -3.0)));
	double tmp;
	if (t_1 <= -5e-22) {
		tmp = t_3;
	} else if (t_1 <= 5e-74) {
		tmp = cos(fma(-0.3333333333333333, (t * z), y)) * t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(2.0 * sqrt(x))
	t_3 = fma(t_2, 1.0, Float64(a / Float64(b * -3.0)))
	tmp = 0.0
	if (t_1 <= -5e-22)
		tmp = t_3;
	elseif (t_1 <= 5e-74)
		tmp = Float64(cos(fma(-0.3333333333333333, Float64(t * z), y)) * t_2);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 1.0 + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-22], t$95$3, If[LessEqual[t$95$1, 5e-74], N[(N[Cos[N[(-0.3333333333333333 * N[(t * z), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -4.99999999999999954e-22 or 4.99999999999999998e-74 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 76.6%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   4. Step-by-step derivation

      1. lower-cos.f64 84.3

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   5. Applied rewrites 84.3%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{a}{b \cdot 3}}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{a}{\color{blue}{b \cdot 3}}\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{\frac{a}{b}}{3}}\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{\color{blue}{\frac{a}{b}}}{3}\right)\right) \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

      14. metadata-eval 84.3

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{\color{blue}{-3}}\right) \]

   7. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{-3}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{-3}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\color{blue}{\frac{a}{b}}}{-3}\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{-3 \cdot b}}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot b}\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\mathsf{neg}\left(3 \cdot b\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{\mathsf{neg}\left(b \cdot 3\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{3 \cdot b}\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot b}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{-3} \cdot b}\right) \]

      13. lower-*.f64 84.3

          \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{-3 \cdot b}}\right) \]

   9. Applied rewrites 84.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{-3 \cdot b}}\right) \]

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, 1, \frac{a}{-3 \cdot b}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 78.1%

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, 1, \frac{a}{-3 \cdot b}\right) \]

   if -4.99999999999999954e-22 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.99999999999999998e-74

   1. Initial program 56.4%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{x} \cdot \left(\cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

   5. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sqrt{x}, 1, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;\frac{a}{b \cdot 3} \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \sqrt{x}, 1, \frac{a}{b \cdot -3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \cos y \cdot \left(2 \cdot \sqrt{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma a (/ -0.3333333333333333 b) (* (cos y) (* 2.0 (sqrt x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(a, (-0.3333333333333333 / b), (cos(y) * (2.0 * sqrt(x))));
}

Julia

function code(x, y, z, t, a, b)
	return fma(a, Float64(-0.3333333333333333 / b), Float64(cos(y) * Float64(2.0 * sqrt(x))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * N[(-0.3333333333333333 / b), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 73.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Applied rewrites 73.6%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3}} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{a}{b \cdot 3}}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{a}{\color{blue}{b \cdot 3}}\right)\right) \]

   10. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{\frac{a}{b}}{3}}\right)\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{\color{blue}{\frac{a}{b}}}{3}\right)\right) \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

   14. metadata-eval 73.6

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{\color{blue}{-3}}\right) \]

7. Applied rewrites 73.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{-3}\right)} \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

      \[\leadsto \color{blue}{\frac{a}{b} \cdot \frac{1}{-3}} + \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]

   5. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a}{b}} \cdot \frac{1}{-3} + \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]

   6. metadata-eval N/A

      \[\leadsto \frac{a}{b} \cdot \color{blue}{\frac{-1}{3}} + \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]

   7. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{a \cdot \frac{-1}{3}}{b}} + \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{a \cdot \frac{\frac{-1}{3}}{b}} + \left(\sqrt{x} \cdot 2\right) \cdot \cos y \]

   9. lift-*.f64 N/A

      \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \color{blue}{\left(\sqrt{x} \cdot 2\right)} \cdot \cos y \]

   10. *-commutative N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \color{blue}{\left(2 \cdot \sqrt{x}\right)} \cdot \cos y \]

   11. lift-sqrt.f64 N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \left(2 \cdot \color{blue}{\sqrt{x}}\right) \cdot \cos y \]

   12. pow1/2 N/A

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

   13. exp-to-pow N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \left(2 \cdot \color{blue}{e^{\log x \cdot \frac{1}{2}}}\right) \cdot \cos y \]

   14. lift-log.f64 N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \left(2 \cdot e^{\color{blue}{\log x} \cdot \frac{1}{2}}\right) \cdot \cos y \]

   15. lift-*.f64 N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \left(2 \cdot e^{\color{blue}{\log x \cdot \frac{1}{2}}}\right) \cdot \cos y \]

   16. lift-exp.f64 N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \left(2 \cdot \color{blue}{e^{\log x \cdot \frac{1}{2}}}\right) \cdot \cos y \]

   17. lift-*.f64 N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \color{blue}{\left(2 \cdot e^{\log x \cdot \frac{1}{2}}\right)} \cdot \cos y \]

   18. lift-*.f64 N/A

       \[\leadsto a \cdot \frac{\frac{-1}{3}}{b} + \color{blue}{\left(2 \cdot e^{\log x \cdot \frac{1}{2}}\right) \cdot \cos y} \]

   19. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\frac{-1}{3}}{b}, \left(2 \cdot e^{\log x \cdot \frac{1}{2}}\right) \cdot \cos y\right)} \]

9. Applied rewrites 73.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \left(2 \cdot \sqrt{x}\right) \cdot \cos y\right)} \]

10. Final simplification 73.5%

    \[\leadsto \mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \cos y \cdot \left(2 \cdot \sqrt{x}\right)\right) \]
11. Add Preprocessing

Alternative 4: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y \cdot 2, \sqrt{x}, \frac{a}{b} \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* (cos y) 2.0) (sqrt x) (* (/ a b) -0.3333333333333333)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((cos(y) * 2.0), sqrt(x), ((a / b) * -0.3333333333333333));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(cos(y) * 2.0), sqrt(x), Float64(Float64(a / b) * -0.3333333333333333))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b}} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\cos y \cdot \sqrt{x}\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \cos y\right) \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{a}{b} \]

   4. metadata-eval N/A

      \[\leadsto \left(2 \cdot \cos y\right) \cdot \sqrt{x} + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 73.4

       \[\leadsto \mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}}\right) \]

5. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \cos y, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\right)} \]

6. Final simplification 73.4%

   \[\leadsto \mathsf{fma}\left(\cos y \cdot 2, \sqrt{x}, \frac{a}{b} \cdot -0.3333333333333333\right) \]
7. Add Preprocessing

Alternative 5: 65.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot \sqrt{x}, 1, \frac{a}{b \cdot -3}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (* 2.0 (sqrt x)) 1.0 (/ a (* b -3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((2.0 * sqrt(x)), 1.0, (a / (b * -3.0)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(2.0 * sqrt(x)), 1.0, Float64(a / Float64(b * -3.0)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 1.0 + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
4. Step-by-step derivation

   1. lower-cos.f64 73.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

5. Applied rewrites 73.6%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3}} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} + \left(\mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \sqrt{x}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot \sqrt{x}}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot 2}, \cos y, \mathsf{neg}\left(\frac{a}{b \cdot 3}\right)\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{a}{b \cdot 3}}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{a}{\color{blue}{b \cdot 3}}\right)\right) \]

   10. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\color{blue}{\frac{\frac{a}{b}}{3}}\right)\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \mathsf{neg}\left(\frac{\color{blue}{\frac{a}{b}}}{3}\right)\right) \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{\mathsf{neg}\left(3\right)}}\right) \]

   14. metadata-eval 73.6

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{\color{blue}{-3}}\right) \]

7. Applied rewrites 73.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\frac{a}{b}}{-3}\right)} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{\frac{a}{b}}{-3}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{\color{blue}{\frac{a}{b}}}{-3}\right) \]

   3. associate-/l/ N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{-3 \cdot b}}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot b}\right) \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\mathsf{neg}\left(3 \cdot b\right)}}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{\mathsf{neg}\left(b \cdot 3\right)}}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{b \cdot 3}\right)}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\mathsf{neg}\left(\color{blue}{3 \cdot b}\right)}\right) \]

   11. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot b}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{-3} \cdot b}\right) \]

   13. lower-*.f64 73.6

       \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \frac{a}{\color{blue}{-3 \cdot b}}\right) \]

9. Applied rewrites 73.6%

   \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, \cos y, \color{blue}{\frac{a}{-3 \cdot b}}\right) \]

10. Taylor expanded in y around 0

    \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, 1, \frac{a}{-3 \cdot b}\right) \]
11. Step-by-step derivation

12. Applied rewrites 58.7%

    \[\leadsto \mathsf{fma}\left(\sqrt{x} \cdot 2, 1, \frac{a}{-3 \cdot b}\right) \]

13. Final simplification 58.7%

    \[\leadsto \mathsf{fma}\left(2 \cdot \sqrt{x}, 1, \frac{a}{b \cdot -3}\right) \]
14. Add Preprocessing

Alternative 6: 50.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b \cdot -3} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ a (* b -3.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a / (b * -3.0);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a / (b * (-3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a / (b * -3.0);
}

Python

def code(x, y, z, t, a, b):
	return a / (b * -3.0)

Julia

function code(x, y, z, t, a, b)
	return Float64(a / Float64(b * -3.0))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a / (b * -3.0);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 44.6

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}} \]

5. Applied rewrites 44.6%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 44.6%

   \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{b}{a}}} \]
8. Step-by-step derivation

9. Applied rewrites 44.8%

   \[\leadsto \frac{a}{\color{blue}{b \cdot -3}} \]
10. Add Preprocessing

Alternative 7: 50.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{b} \cdot a \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (/ -0.3333333333333333 b) a))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (-0.3333333333333333 / b) * a;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((-0.3333333333333333d0) / b) * a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (-0.3333333333333333 / b) * a;
}

Python

def code(x, y, z, t, a, b):
	return (-0.3333333333333333 / b) * a

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(-0.3333333333333333 / b) * a)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (-0.3333333333333333 / b) * a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(-0.3333333333333333 / b), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 44.6

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}} \]

5. Applied rewrites 44.6%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
6. Step-by-step derivation

7. Applied rewrites 44.6%

   \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{b}{a}}} \]
8. Step-by-step derivation

9. Applied rewrites 44.7%

   \[\leadsto \frac{-0.3333333333333333}{b} \cdot \color{blue}{a} \]
10. Add Preprocessing

Alternative 8: 50.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (/ a b) -0.3333333333333333))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (a / b) * -0.3333333333333333;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a / b) * (-0.3333333333333333d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (a / b) * -0.3333333333333333;
}

Python

def code(x, y, z, t, a, b):
	return (a / b) * -0.3333333333333333

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(a / b) * -0.3333333333333333)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (a / b) * -0.3333333333333333;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 44.6

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{a}{b}} \]

5. Applied rewrites 44.6%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]

6. Final simplification 44.6%

   \[\leadsto \frac{a}{b} \cdot -0.3333333333333333 \]
7. Add Preprocessing

Developer Target 1: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t\_3 \cdot \cos \left(\frac{1}{y} - t\_1\right) - t\_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t\_1\right) \cdot t\_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024296 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -1379333748723514100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 3333333333333333/10000000000000000 z) t)))) (/ (/ a 3) b)) (if (< z 35162906135559870000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 3333333333333333/10000000000000000 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3)))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))