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

Percentage Accurate: 70.0% → 76.6%

Time: 21.5s

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: 70.0% 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: 76.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in y around inf

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

5. Simplified 78.3%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 78.4

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

7. Applied egg-rr 78.4%

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

Alternative 2: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-127}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{fma}\left(t, -0.3333333333333333 \cdot z, y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (fma (sqrt x) 2.0 (/ a (* b -3.0)))))
   (if (<= t_1 -2e-60)
     t_2
     (if (<= t_1 4e-127)
       (* 2.0 (* (sqrt x) (cos (fma t (* -0.3333333333333333 z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = fma(sqrt(x), 2.0, (a / (b * -3.0)));
	double tmp;
	if (t_1 <= -2e-60) {
		tmp = t_2;
	} else if (t_1 <= 4e-127) {
		tmp = 2.0 * (sqrt(x) * cos(fma(t, (-0.3333333333333333 * z), y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = fma(sqrt(x), 2.0, Float64(a / Float64(b * -3.0)))
	tmp = 0.0
	if (t_1 <= -2e-60)
		tmp = t_2;
	elseif (t_1 <= 4e-127)
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(fma(t, Float64(-0.3333333333333333 * z), y))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-60], t$95$2, If[LessEqual[t$95$1, 4e-127], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(t * N[(-0.3333333333333333 * z), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.9999999999999999e-60 or 4.0000000000000001e-127 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 90.1%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval 90.1

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

   7. Applied egg-rr 90.1%

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

   8. Taylor expanded in y around 0

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

      1. sqrt-lowering-sqrt.f64 83.2

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

   10. Simplified 83.2%

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

   if -1.9999999999999999e-60 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.0000000000000001e-127

   1. Initial program 57.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 55.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right), 2 \cdot \sqrt{x}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. rem-exp-log N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 55.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 4 \cdot 10^{-127}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{fma}\left(t, -0.3333333333333333 \cdot z, y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (fma (sqrt x) 2.0 (/ a (* b -3.0)))))
   (if (<= t_1 -2e-60)
     t_2
     (if (<= t_1 4e-127)
       (* (* 2.0 (sqrt x)) (cos (fma -0.3333333333333333 (* t z) y)))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = fma(sqrt(x), 2.0, (a / (b * -3.0)));
	double tmp;
	if (t_1 <= -2e-60) {
		tmp = t_2;
	} else if (t_1 <= 4e-127) {
		tmp = (2.0 * sqrt(x)) * cos(fma(-0.3333333333333333, (t * z), y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = fma(sqrt(x), 2.0, Float64(a / Float64(b * -3.0)))
	tmp = 0.0
	if (t_1 <= -2e-60)
		tmp = t_2;
	elseif (t_1 <= 4e-127)
		tmp = Float64(Float64(2.0 * sqrt(x)) * cos(fma(-0.3333333333333333, Float64(t * z), y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-60], t$95$2, If[LessEqual[t$95$1, 4e-127], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.3333333333333333 * N[(t * z), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.9999999999999999e-60 or 4.0000000000000001e-127 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 90.1%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval 90.1

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

   7. Applied egg-rr 90.1%

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

   8. Taylor expanded in y around 0

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

      1. sqrt-lowering-sqrt.f64 83.2

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

   10. Simplified 83.2%

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

   if -1.9999999999999999e-60 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.0000000000000001e-127

   1. Initial program 57.1%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 57.1

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

   4. Applied egg-rr 57.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)} \]
   6. 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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

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

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. *-lowering-*.f64 55.6

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

   7. Simplified 55.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\mathsf{fma}\left(-0.3333333333333333, t \cdot z, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (fma (sqrt x) 2.0 (/ a (* b -3.0)))))
   (if (<= t_1 -2e-60)
     t_2
     (if (<= t_1 4e-127) (* (* 2.0 (sqrt x)) (cos y)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = fma(sqrt(x), 2.0, (a / (b * -3.0)));
	double tmp;
	if (t_1 <= -2e-60) {
		tmp = t_2;
	} else if (t_1 <= 4e-127) {
		tmp = (2.0 * sqrt(x)) * cos(y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = fma(sqrt(x), 2.0, Float64(a / Float64(b * -3.0)))
	tmp = 0.0
	if (t_1 <= -2e-60)
		tmp = t_2;
	elseif (t_1 <= 4e-127)
		tmp = Float64(Float64(2.0 * sqrt(x)) * cos(y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * 2.0 + N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-60], t$95$2, If[LessEqual[t$95$1, 4e-127], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.9999999999999999e-60 or 4.0000000000000001e-127 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 90.1%

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

      1. sub-neg N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. distribute-neg-frac2 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. metadata-eval 90.1

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

   7. Applied egg-rr 90.1%

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

   8. Taylor expanded in y around 0

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

      1. sqrt-lowering-sqrt.f64 83.2

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

   10. Simplified 83.2%

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

   if -1.9999999999999999e-60 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.0000000000000001e-127

   1. Initial program 57.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 56.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\cos y \cdot \sqrt{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos y} \cdot \left(2 \cdot \sqrt{x}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} \]

      8. sqrt-lowering-sqrt.f64 55.0

         \[\leadsto \cos y \cdot \left(2 \cdot \color{blue}{\sqrt{x}}\right) \]

   8. Simplified 55.0%

      \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, 2, \frac{a}{b \cdot -3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in y around inf

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

5. Simplified 78.3%

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

   1. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. *-lowering-*.f64 N/A

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

   12. metadata-eval 78.3

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

7. Applied egg-rr 78.3%

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

Alternative 6: 76.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in y around inf

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

5. Simplified 78.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right) - \frac{1}{3} \cdot \frac{a}{b}} \]
7. 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. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*r/ N/A

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

   11. /-lowering-/.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 78.3

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

8. Simplified 78.3%

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

Alternative 7: 64.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in y around inf

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

5. Simplified 78.3%

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

   1. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. distribute-neg-frac2 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. *-lowering-*.f64 N/A

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

   12. metadata-eval 78.3

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

7. Applied egg-rr 78.3%

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

8. Taylor expanded in y around 0

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

   1. sqrt-lowering-sqrt.f64 67.3

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

10. Simplified 67.3%

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

Alternative 8: 64.7% accurate, 4.8× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Taylor expanded in y around inf

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

5. Simplified 78.3%

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

6. Taylor expanded in y around 0

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

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

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \sqrt{x} + \color{blue}{\frac{-1}{3}} \cdot \frac{a}{b} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b} + 2 \cdot \sqrt{x}} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot a}{b}} + 2 \cdot \sqrt{x} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{a \cdot \frac{-1}{3}}}{b} + 2 \cdot \sqrt{x} \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{a \cdot \frac{\frac{-1}{3}}{b}} + 2 \cdot \sqrt{x} \]

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f64 N/A

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

   17. *-lowering-*.f64 N/A

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

   18. sqrt-lowering-sqrt.f64 67.3

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

8. Simplified 67.3%

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

Alternative 9: 50.3% 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 71.1%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. /-lowering-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 51.3

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

5. Simplified 51.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 51.2

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

7. Applied egg-rr 51.2%

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

   1. metadata-eval N/A

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

   2. times-frac N/A

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

   3. *-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-neg-frac N/A

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

   6. distribute-neg-frac2 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. *-lowering-*.f64 N/A

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

   10. metadata-eval 51.4

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

9. Applied egg-rr 51.4%

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

Alternative 10: 50.2% accurate, 9.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return a * (-0.3333333333333333 / 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 * ((-0.3333333333333333d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. /-lowering-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 51.3

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

5. Simplified 51.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 51.3

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

7. Applied egg-rr 51.3%

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

8. Final simplification 51.3%

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

Alternative 11: 50.2% accurate, 9.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return -0.3333333333333333 * (a / 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 = (-0.3333333333333333d0) * (a / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. /-lowering-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 51.3

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

5. Simplified 51.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 51.2

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

7. Applied egg-rr 51.2%

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

Developer Target 1: 74.1% 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 2024195 
(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))))