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

Percentage Accurate: 70.1% → 76.8%

Time: 16.7s

Alternatives: 11

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: 70.1% 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: 76.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((sqrt(x) * 2.0) * cos(y)) - (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 = ((sqrt(x) * 2.0d0) * cos(y)) - (a / (b * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in t 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 77.9

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

5. Applied rewrites 77.9%

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

6. Final simplification 77.9%

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

Alternative 2: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(t \cdot z, -0.3333333333333333, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (fma (sqrt x) 2.0 (* (/ a b) -0.3333333333333333))))
   (if (<= t_1 -5e-66)
     t_2
     (if (<= t_1 5e-141)
       (* (cos (fma (* t z) -0.3333333333333333 y)) (* (sqrt x) 2.0))
       t_2))))

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 = fma(sqrt(x), 2.0, ((a / b) * -0.3333333333333333));
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_2;
	} else if (t_1 <= 5e-141) {
		tmp = cos(fma((t * z), -0.3333333333333333, y)) * (sqrt(x) * 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = fma(sqrt(x), 2.0, Float64(Float64(a / b) * -0.3333333333333333))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_2;
	elseif (t_1 <= 5e-141)
		tmp = Float64(cos(fma(Float64(t * z), -0.3333333333333333, y)) * Float64(sqrt(x) * 2.0));
	else
		tmp = t_2;
	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[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-66], t$95$2, If[LessEqual[t$95$1, 5e-141], N[(N[Cos[N[(N[(t * z), $MachinePrecision] * -0.3333333333333333 + y), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -4.99999999999999962e-66 or 4.9999999999999999e-141 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 58.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(y, y, {\left(t \cdot z\right)}^{2} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right)} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left({\left(t \cdot z\right)}^{2}\right)}^{1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      2. pow-to-exp N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      3. lower-exp.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-log.f64 58.8

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot 1} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   6. Applied rewrites 58.8%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{1 \cdot \log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      4. exp-prod N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      6. exp-1-e N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      7. lower-E.f64 58.9

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   8. Applied rewrites 58.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\mathsf{E}\left(\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

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

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

       6. metadata-eval N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 85.0

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

   11. Applied rewrites 85.0%

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

   if -4.99999999999999962e-66 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.9999999999999999e-141

   1. Initial program 50.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{b} \cdot a \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{b}\right)\right)} \cdot a \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{b}\right)\right) \cdot a \]

      6. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{b}}\right)\right) \cdot a \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{b}\right)\right) \cdot a} \]

      8. associate-*r/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{b}}\right)\right) \cdot a \]

      9. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{b}\right)\right) \cdot a \]

      10. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{b}} \cdot a \]

      11. metadata-eval N/A

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

      12. lower-/.f64 3.5

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

   5. Applied rewrites 3.5%

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

   7. Applied rewrites 3.5%

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

   8. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 51.3

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

   10. Applied rewrites 51.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;\frac{a}{b \cdot 3} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(t \cdot z, -0.3333333333333333, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot z, t, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (fma (sqrt x) 2.0 (* (/ a b) -0.3333333333333333))))
   (if (<= t_1 -5e-66)
     t_2
     (if (<= t_1 5e-141)
       (* (cos (fma (* -0.3333333333333333 z) t y)) (* (sqrt x) 2.0))
       t_2))))

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 = fma(sqrt(x), 2.0, ((a / b) * -0.3333333333333333));
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_2;
	} else if (t_1 <= 5e-141) {
		tmp = cos(fma((-0.3333333333333333 * z), t, y)) * (sqrt(x) * 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = fma(sqrt(x), 2.0, Float64(Float64(a / b) * -0.3333333333333333))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_2;
	elseif (t_1 <= 5e-141)
		tmp = Float64(cos(fma(Float64(-0.3333333333333333 * z), t, y)) * Float64(sqrt(x) * 2.0));
	else
		tmp = t_2;
	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[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-66], t$95$2, If[LessEqual[t$95$1, 5e-141], N[(N[Cos[N[(N[(-0.3333333333333333 * z), $MachinePrecision] * t + y), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -4.99999999999999962e-66 or 4.9999999999999999e-141 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 58.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(y, y, {\left(t \cdot z\right)}^{2} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right)} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left({\left(t \cdot z\right)}^{2}\right)}^{1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      2. pow-to-exp N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      3. lower-exp.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-log.f64 58.8

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot 1} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   6. Applied rewrites 58.8%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{1 \cdot \log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      4. exp-prod N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      6. exp-1-e N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      7. lower-E.f64 58.9

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   8. Applied rewrites 58.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\mathsf{E}\left(\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

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

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

       6. metadata-eval N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 85.0

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

   11. Applied rewrites 85.0%

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

   if -4.99999999999999962e-66 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.9999999999999999e-141

   1. Initial program 50.8%

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

   3. Taylor expanded in t 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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. distribute-neg-in N/A

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

      13. lower-cos.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

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

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

      17. metadata-eval N/A

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

   8. Applied rewrites 51.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;\frac{a}{b \cdot 3} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot z, t, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, z, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0)))
        (t_2 (fma (sqrt x) 2.0 (* (/ a b) -0.3333333333333333))))
   (if (<= t_1 -5e-66)
     t_2
     (if (<= t_1 5e-141)
       (* (cos (fma (* -0.3333333333333333 t) z y)) (* (sqrt x) 2.0))
       t_2))))

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 = fma(sqrt(x), 2.0, ((a / b) * -0.3333333333333333));
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_2;
	} else if (t_1 <= 5e-141) {
		tmp = cos(fma((-0.3333333333333333 * t), z, y)) * (sqrt(x) * 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = fma(sqrt(x), 2.0, Float64(Float64(a / b) * -0.3333333333333333))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_2;
	elseif (t_1 <= 5e-141)
		tmp = Float64(cos(fma(Float64(-0.3333333333333333 * t), z, y)) * Float64(sqrt(x) * 2.0));
	else
		tmp = t_2;
	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[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-66], t$95$2, If[LessEqual[t$95$1, 5e-141], N[(N[Cos[N[(N[(-0.3333333333333333 * t), $MachinePrecision] * z + y), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -4.99999999999999962e-66 or 4.9999999999999999e-141 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 58.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(y, y, {\left(t \cdot z\right)}^{2} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right)} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left({\left(t \cdot z\right)}^{2}\right)}^{1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      2. pow-to-exp N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      3. lower-exp.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-log.f64 58.8

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot 1} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   6. Applied rewrites 58.8%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{1 \cdot \log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      4. exp-prod N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      6. exp-1-e N/A

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

      7. lower-E.f64 58.9

         \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   8. Applied rewrites 58.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\mathsf{E}\left(\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

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

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

       6. metadata-eval N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 85.0

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

   11. Applied rewrites 85.0%

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

   if -4.99999999999999962e-66 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.9999999999999999e-141

   1. Initial program 50.8%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 50.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;\frac{a}{b \cdot 3} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, z, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b} \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in t 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 77.9

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

5. Applied rewrites 77.9%

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

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

7. Applied rewrites 77.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 77.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 77.8

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

9. Applied rewrites 77.8%

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

10. Final simplification 77.8%

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

Alternative 6: 76.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in t 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. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 77.8

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

5. Applied rewrites 77.8%

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

6. Final simplification 77.8%

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

Alternative 7: 64.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

4. Applied rewrites 49.6%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(y, y, {\left(t \cdot z\right)}^{2} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right)} - \frac{a}{b \cdot 3} \]
5. Step-by-step derivation

   1. unpow1 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left({\left(t \cdot z\right)}^{2}\right)}^{1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   2. pow-to-exp N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   3. lower-exp.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   5. lower-log.f64 49.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot 1} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

6. Applied rewrites 49.7%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]
7. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{e^{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{\log \left({\left(t \cdot z\right)}^{2}\right) \cdot 1}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   3. *-commutative N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, e^{\color{blue}{1 \cdot \log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   4. exp-prod N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\left(e^{1}\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   6. exp-1-e N/A

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot \frac{-1}{9}\right)}{\mathsf{fma}\left(\frac{1}{3}, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

   7. lower-E.f64 49.9

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, {\color{blue}{\mathsf{E}\left(\right)}}^{\log \left({\left(t \cdot z\right)}^{2}\right)} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

8. Applied rewrites 49.9%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(y, y, \color{blue}{{\mathsf{E}\left(\right)}^{\log \left({\left(t \cdot z\right)}^{2}\right)}} \cdot -0.1111111111111111\right)}{\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)}\right) - \frac{a}{b \cdot 3} \]

9. Taylor expanded in t around inf

   \[\leadsto \color{blue}{2 \cdot \sqrt{x} - \frac{1}{3} \cdot \frac{a}{b}} \]
10. Step-by-step derivation

    1. sub-neg N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-sqrt.f64 N/A

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

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

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

    6. metadata-eval N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 64.7

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

11. Applied rewrites 64.7%

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

Alternative 8: 49.8% accurate, 6.9× speedup

Math

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

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(Float64(a / 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[(N[(a / b), $MachinePrecision] / -3.0), $MachinePrecision]

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{b} \cdot a \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{b}\right)\right)} \cdot a \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{b}\right)\right) \cdot a \]

   6. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{b}}\right)\right) \cdot a \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{b}\right)\right) \cdot a} \]

   8. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{b}}\right)\right) \cdot a \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{b}\right)\right) \cdot a \]

   10. distribute-neg-frac N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{b}} \cdot a \]

   11. metadata-eval N/A

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

   12. lower-/.f64 51.5

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

5. Applied rewrites 51.5%

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

7. Applied rewrites 51.5%

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

Alternative 9: 49.8% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

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 / ((-3.0d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{b} \cdot a \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{b}\right)\right)} \cdot a \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{b}\right)\right) \cdot a \]

   6. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{b}}\right)\right) \cdot a \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{b}\right)\right) \cdot a} \]

   8. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{b}}\right)\right) \cdot a \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{b}\right)\right) \cdot a \]

   10. distribute-neg-frac N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{b}} \cdot a \]

   11. metadata-eval N/A

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

   12. lower-/.f64 51.5

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

5. Applied rewrites 51.5%

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

7. Applied rewrites 51.5%

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

Alternative 10: 49.7% 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 72.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{b} \cdot a \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{b}\right)\right)} \cdot a \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{b}\right)\right) \cdot a \]

   6. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{b}}\right)\right) \cdot a \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{b}\right)\right) \cdot a} \]

   8. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{b}}\right)\right) \cdot a \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{b}\right)\right) \cdot a \]

   10. distribute-neg-frac N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{b}} \cdot a \]

   11. metadata-eval N/A

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

   12. lower-/.f64 51.5

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

5. Applied rewrites 51.5%

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

7. Applied rewrites 51.5%

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

Alternative 11: 49.7% 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 72.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{b} \cdot a \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3}}{b}\right)\right)} \cdot a \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3} \cdot 1}}{b}\right)\right) \cdot a \]

   6. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{1}{b}}\right)\right) \cdot a \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{b}\right)\right) \cdot a} \]

   8. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{b}}\right)\right) \cdot a \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{b}\right)\right) \cdot a \]

   10. distribute-neg-frac N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{b}} \cdot a \]

   11. metadata-eval N/A

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

   12. lower-/.f64 51.5

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

5. Applied rewrites 51.5%

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

Developer Target 1: 74.2% 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 2024273 
(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))))