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

Percentage Accurate: 70.6% → 77.2%

Time: 20.6s

Alternatives: 10

Speedup: 1.0×

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: 70.6% 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: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 1:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \frac{-1}{\frac{3}{z \cdot t}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \mathsf{expm1}\left(\log 2 + {y}^{2} \cdot -0.25\right)\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 1.0)
     (- (* (* 2.0 (sqrt x)) (cos (+ y (/ -1.0 (/ 3.0 (* z t)))))) t_1)
     (-
      (* 2.0 (* (sqrt x) (expm1 (+ (log 2.0) (* (pow y 2.0) -0.25)))))
      t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = ((2.0 * sqrt(x)) * cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * expm1((log(2.0) + (pow(y, 2.0) * -0.25))))) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = ((2.0 * Math.sqrt(x)) * Math.cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * Math.expm1((Math.log(2.0) + (Math.pow(y, 2.0) * -0.25))))) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 1.0:
		tmp = ((2.0 * math.sqrt(x)) * math.cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1
	else:
		tmp = (2.0 * (math.sqrt(x) * math.expm1((math.log(2.0) + (math.pow(y, 2.0) * -0.25))))) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 1.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y + Float64(-1.0 / Float64(3.0 / Float64(z * t)))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * expm1(Float64(log(2.0) + Float64((y ^ 2.0) * -0.25))))) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y + N[(-1.0 / N[(3.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(Exp[N[(N[Log[2.0], $MachinePrecision] + N[(N[Power[y, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 1

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-/l* 79.4%

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

      6. *-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 79.9%

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

      2. clear-num 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 1 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 53.2%

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

      1. expm1-log1p-u 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

      2. expm1-undefine 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]

   5. Applied egg-rr 53.2%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

      1. expm1-define 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

   7. Simplified 53.2%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

   8. Taylor expanded in y around 0 54.0%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \mathsf{expm1}\left(\color{blue}{\log 2 + -0.25 \cdot {y}^{2}}\right)\right) - \frac{a}{b \cdot 3} \]
   9. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \mathsf{expm1}\left(\log 2 + \color{blue}{{y}^{2} \cdot -0.25}\right)\right) - \frac{a}{b \cdot 3} \]

   10. Simplified 54.0%

       \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \mathsf{expm1}\left(\color{blue}{\log 2 + {y}^{2} \cdot -0.25}\right)\right) - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 1:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \frac{-1}{\frac{3}{z \cdot t}}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \mathsf{expm1}\left(\log 2 + {y}^{2} \cdot -0.25\right)\right) - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - t\_1\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\left(1 + \cos y\right) + -1\right)\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b)))
        (t_2 (- (* (cos (- y (/ (* z t) 3.0))) (* 2.0 (sqrt x))) t_1)))
   (if (<= t_2 2e+144)
     t_2
     (- (* 2.0 (* (sqrt x) (+ (+ 1.0 (cos y)) -1.0))) t_1))))

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 = (cos((y - ((z * t) / 3.0))) * (2.0 * sqrt(x))) - t_1;
	double tmp;
	if (t_2 <= 2e+144) {
		tmp = t_2;
	} else {
		tmp = (2.0 * (sqrt(x) * ((1.0 + cos(y)) + -1.0))) - t_1;
	}
	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) :: tmp
    t_1 = a / (3.0d0 * b)
    t_2 = (cos((y - ((z * t) / 3.0d0))) * (2.0d0 * sqrt(x))) - t_1
    if (t_2 <= 2d+144) then
        tmp = t_2
    else
        tmp = (2.0d0 * (sqrt(x) * ((1.0d0 + cos(y)) + (-1.0d0)))) - t_1
    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 = a / (3.0 * b);
	double t_2 = (Math.cos((y - ((z * t) / 3.0))) * (2.0 * Math.sqrt(x))) - t_1;
	double tmp;
	if (t_2 <= 2e+144) {
		tmp = t_2;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * ((1.0 + Math.cos(y)) + -1.0))) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = (math.cos((y - ((z * t) / 3.0))) * (2.0 * math.sqrt(x))) - t_1
	tmp = 0
	if t_2 <= 2e+144:
		tmp = t_2
	else:
		tmp = (2.0 * (math.sqrt(x) * ((1.0 + math.cos(y)) + -1.0))) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(Float64(cos(Float64(y - Float64(Float64(z * t) / 3.0))) * Float64(2.0 * sqrt(x))) - t_1)
	tmp = 0.0
	if (t_2 <= 2e+144)
		tmp = t_2;
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(1.0 + cos(y)) + -1.0))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = (cos((y - ((z * t) / 3.0))) * (2.0 * sqrt(x))) - t_1;
	tmp = 0.0;
	if (t_2 <= 2e+144)
		tmp = t_2;
	else
		tmp = (2.0 * (sqrt(x) * ((1.0 + cos(y)) + -1.0))) - t_1;
	end
	tmp_2 = 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[(N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 2e+144], t$95$2, N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(1.0 + N[Cos[y], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64)))) < 2.00000000000000005e144

   1. Initial program 77.2%

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

   if 2.00000000000000005e144 < (-.f64 (*.f64 (*.f64 #s(literal 2 binary64) (sqrt.f64 x)) (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))) (/.f64 a (*.f64 b #s(literal 3 binary64))))

   1. Initial program 48.6%

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

   3. Taylor expanded in z around 0 75.3%

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

      1. expm1-log1p-u 75.3%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

      2. expm1-undefine 75.3%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]

   5. Applied egg-rr 75.3%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

      1. expm1-define 75.3%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

   7. Simplified 75.3%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]
   8. Step-by-step derivation

      1. expm1-undefine 75.3%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]

      2. log1p-undefine 75.3%

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

      3. rem-exp-log 75.3%

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

   9. Applied egg-rr 75.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{3 \cdot b} \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\cos \left(y - \frac{z \cdot t}{3}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\left(1 + \cos y\right) + -1\right)\right) - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \frac{-1}{\frac{3}{z \cdot t}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(e^{\log x \cdot 0.5} \cdot \cos y\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 2.0)
     (- (* (* 2.0 (sqrt x)) (cos (+ y (/ -1.0 (/ 3.0 (* z t)))))) t_1)
     (- (* 2.0 (* (exp (* (log x) 0.5)) (cos y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * sqrt(x)) * cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	} else {
		tmp = (2.0 * (exp((log(x) * 0.5)) * cos(y))) - t_1;
	}
	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) :: tmp
    t_1 = a / (3.0d0 * b)
    if (cos((y - ((z * t) / 3.0d0))) <= 2.0d0) then
        tmp = ((2.0d0 * sqrt(x)) * cos((y + ((-1.0d0) / (3.0d0 / (z * t)))))) - t_1
    else
        tmp = (2.0d0 * (exp((log(x) * 0.5d0)) * cos(y))) - t_1
    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 = a / (3.0 * b);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 2.0) {
		tmp = ((2.0 * Math.sqrt(x)) * Math.cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	} else {
		tmp = (2.0 * (Math.exp((Math.log(x) * 0.5)) * Math.cos(y))) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 2.0:
		tmp = ((2.0 * math.sqrt(x)) * math.cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1
	else:
		tmp = (2.0 * (math.exp((math.log(x) * 0.5)) * math.cos(y))) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 2.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y + Float64(-1.0 / Float64(3.0 / Float64(z * t)))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(exp(Float64(log(x) * 0.5)) * cos(y))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 2.0)
		tmp = ((2.0 * sqrt(x)) * cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	else
		tmp = (2.0 * (exp((log(x) * 0.5)) * cos(y))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y + N[(-1.0 / N[(3.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(2.0 * N[(N[Exp[N[(N[Log[x], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 2

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-/l* 79.4%

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

      6. *-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 79.9%

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

      2. clear-num 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 2 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 53.2%

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

      1. pow1/2 53.2%

         \[\leadsto 2 \cdot \left(\color{blue}{{x}^{0.5}} \cdot \cos y\right) - \frac{a}{b \cdot 3} \]

      2. pow-to-exp 53.2%

         \[\leadsto 2 \cdot \left(\color{blue}{e^{\log x \cdot 0.5}} \cdot \cos y\right) - \frac{a}{b \cdot 3} \]

   5. Applied egg-rr 53.2%

      \[\leadsto 2 \cdot \left(\color{blue}{e^{\log x \cdot 0.5}} \cdot \cos y\right) - \frac{a}{b \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 2:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \frac{-1}{\frac{3}{z \cdot t}}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(e^{\log x \cdot 0.5} \cdot \cos y\right) - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 1:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \frac{-1}{\frac{3}{z \cdot t}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\left(1 + \cos y\right) + -1\right)\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= (cos (- y (/ (* z t) 3.0))) 1.0)
     (- (* (* 2.0 (sqrt x)) (cos (+ y (/ -1.0 (/ 3.0 (* z t)))))) t_1)
     (- (* 2.0 (* (sqrt x) (+ (+ 1.0 (cos y)) -1.0))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if (cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = ((2.0 * sqrt(x)) * cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	} else {
		tmp = (2.0 * (sqrt(x) * ((1.0 + cos(y)) + -1.0))) - t_1;
	}
	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) :: tmp
    t_1 = a / (3.0d0 * b)
    if (cos((y - ((z * t) / 3.0d0))) <= 1.0d0) then
        tmp = ((2.0d0 * sqrt(x)) * cos((y + ((-1.0d0) / (3.0d0 / (z * t)))))) - t_1
    else
        tmp = (2.0d0 * (sqrt(x) * ((1.0d0 + cos(y)) + (-1.0d0)))) - t_1
    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 = a / (3.0 * b);
	double tmp;
	if (Math.cos((y - ((z * t) / 3.0))) <= 1.0) {
		tmp = ((2.0 * Math.sqrt(x)) * Math.cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	} else {
		tmp = (2.0 * (Math.sqrt(x) * ((1.0 + Math.cos(y)) + -1.0))) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if math.cos((y - ((z * t) / 3.0))) <= 1.0:
		tmp = ((2.0 * math.sqrt(x)) * math.cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1
	else:
		tmp = (2.0 * (math.sqrt(x) * ((1.0 + math.cos(y)) + -1.0))) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (cos(Float64(y - Float64(Float64(z * t) / 3.0))) <= 1.0)
		tmp = Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y + Float64(-1.0 / Float64(3.0 / Float64(z * t)))))) - t_1);
	else
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(1.0 + cos(y)) + -1.0))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if (cos((y - ((z * t) / 3.0))) <= 1.0)
		tmp = ((2.0 * sqrt(x)) * cos((y + (-1.0 / (3.0 / (z * t)))))) - t_1;
	else
		tmp = (2.0 * (sqrt(x) * ((1.0 + cos(y)) + -1.0))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64)))) < 1

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-/l* 79.4%

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

      6. *-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 79.9%

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

      2. clear-num 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 1 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) #s(literal 3 binary64))))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 53.2%

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

      1. expm1-log1p-u 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

      2. expm1-undefine 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]

   5. Applied egg-rr 53.2%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]
   6. Step-by-step derivation

      1. expm1-define 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]

   7. Simplified 53.2%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)}\right) - \frac{a}{b \cdot 3} \]
   8. Step-by-step derivation

      1. expm1-undefine 53.2%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos y\right)} - 1\right)}\right) - \frac{a}{b \cdot 3} \]

      2. log1p-undefine 53.2%

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

      3. rem-exp-log 53.2%

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

   9. Applied egg-rr 53.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 1:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \frac{-1}{\frac{3}{z \cdot t}}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\left(1 + \cos y\right) + -1\right)\right) - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-107} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-142}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y + -0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (or (<= t_1 -2e-107) (not (<= t_1 2e-142)))
     (- (* 2.0 (sqrt x)) t_1)
     (* 2.0 (* (sqrt x) (cos (+ y (* -0.3333333333333333 (* z t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -2e-107) || !(t_1 <= 2e-142)) {
		tmp = (2.0 * sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (sqrt(x) * cos((y + (-0.3333333333333333 * (z * t)))));
	}
	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) :: tmp
    t_1 = a / (3.0d0 * b)
    if ((t_1 <= (-2d-107)) .or. (.not. (t_1 <= 2d-142))) then
        tmp = (2.0d0 * sqrt(x)) - t_1
    else
        tmp = 2.0d0 * (sqrt(x) * cos((y + ((-0.3333333333333333d0) * (z * t)))))
    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 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -2e-107) || !(t_1 <= 2e-142)) {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (Math.sqrt(x) * Math.cos((y + (-0.3333333333333333 * (z * t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if (t_1 <= -2e-107) or not (t_1 <= 2e-142):
		tmp = (2.0 * math.sqrt(x)) - t_1
	else:
		tmp = 2.0 * (math.sqrt(x) * math.cos((y + (-0.3333333333333333 * (z * t)))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_1 <= -2e-107) || !(t_1 <= 2e-142))
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_1);
	else
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(Float64(y + Float64(-0.3333333333333333 * Float64(z * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_1 <= -2e-107) || ~((t_1 <= 2e-142)))
		tmp = (2.0 * sqrt(x)) - t_1;
	else
		tmp = 2.0 * (sqrt(x) * cos((y + (-0.3333333333333333 * (z * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-107], N[Not[LessEqual[t$95$1, 2e-142]], $MachinePrecision]], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(y + N[(-0.3333333333333333 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2e-107 or 2.0000000000000001e-142 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 77.1%

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

   3. Taylor expanded in z around 0 85.3%

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

   4. Taylor expanded in y around 0 81.4%

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

      1. *-commutative 81.4%

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

   6. Simplified 81.4%

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

   if -2e-107 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-142

   1. Initial program 57.6%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around inf 56.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-107} \lor \neg \left(\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-142}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y + -0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.0× speedup

Math

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

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(2.0 * Float64(sqrt(x) * cos(y))) - Float64(a / Float64(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[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in z around 0 74.9%

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

4. Final simplification 74.9%

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

Alternative 7: 60.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-49} \lor \neg \left(t\_1 \leq 10^{-124}\right):\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} + 0.3333333333333333 \cdot \frac{a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (or (<= t_1 -2e-49) (not (<= t_1 1e-124)))
     (/ a (* b -3.0))
     (+ (* 2.0 (sqrt x)) (* 0.3333333333333333 (/ a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -2e-49) || !(t_1 <= 1e-124)) {
		tmp = a / (b * -3.0);
	} else {
		tmp = (2.0 * sqrt(x)) + (0.3333333333333333 * (a / b));
	}
	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) :: tmp
    t_1 = a / (3.0d0 * b)
    if ((t_1 <= (-2d-49)) .or. (.not. (t_1 <= 1d-124))) then
        tmp = a / (b * (-3.0d0))
    else
        tmp = (2.0d0 * sqrt(x)) + (0.3333333333333333d0 * (a / b))
    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 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -2e-49) || !(t_1 <= 1e-124)) {
		tmp = a / (b * -3.0);
	} else {
		tmp = (2.0 * Math.sqrt(x)) + (0.3333333333333333 * (a / b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if (t_1 <= -2e-49) or not (t_1 <= 1e-124):
		tmp = a / (b * -3.0)
	else:
		tmp = (2.0 * math.sqrt(x)) + (0.3333333333333333 * (a / b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_1 <= -2e-49) || !(t_1 <= 1e-124))
		tmp = Float64(a / Float64(b * -3.0));
	else
		tmp = Float64(Float64(2.0 * sqrt(x)) + Float64(0.3333333333333333 * Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_1 <= -2e-49) || ~((t_1 <= 1e-124)))
		tmp = a / (b * -3.0);
	else
		tmp = (2.0 * sqrt(x)) + (0.3333333333333333 * (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-49], N[Not[LessEqual[t$95$1, 1e-124]], $MachinePrecision]], N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.99999999999999987e-49 or 9.99999999999999933e-125 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 79.2%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in a around inf 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l/ 80.3%

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

      3. associate-*r/ 80.3%

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

      4. clear-num 80.2%

         \[\leadsto a \cdot \color{blue}{\frac{1}{\frac{b}{-0.3333333333333333}}} \]

      5. un-div-inv 80.3%

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

      6. div-inv 80.4%

         \[\leadsto \frac{a}{\color{blue}{b \cdot \frac{1}{-0.3333333333333333}}} \]

      7. metadata-eval 80.4%

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

      8. metadata-eval 80.4%

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

      9. distribute-rgt-neg-in 80.4%

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

      10. *-commutative 80.4%

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

      11. distribute-lft-neg-in 80.4%

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

      12. metadata-eval 80.4%

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

   7. Applied egg-rr 80.4%

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

   if -1.99999999999999987e-49 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 9.99999999999999933e-125

   1. Initial program 56.8%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in y around 0 30.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b} + 2 \cdot \left(\sqrt{x} \cdot \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) + -0.3333333333333333 \cdot \frac{a}{b}} \]

      2. associate-*r/ 30.6%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) + \color{blue}{\frac{-0.3333333333333333 \cdot a}{b}} \]

      3. *-commutative 30.6%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) + \frac{\color{blue}{a \cdot -0.3333333333333333}}{b} \]

      4. associate-*r/ 30.7%

         \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \cos \left(-0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right) + \color{blue}{a \cdot \frac{-0.3333333333333333}{b}} \]

      5. fma-define 30.7%

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

      6. metadata-eval 30.7%

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

      7. distribute-lft-neg-in 30.7%

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

      8. cos-neg 30.7%

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

      9. rem-square-sqrt 23.9%

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

      10. fabs-sqr 23.9%

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

      11. rem-square-sqrt 30.7%

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

      12. associate-*r/ 30.6%

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

      13. *-commutative 30.6%

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

      14. associate-*r/ 30.6%

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

      15. metadata-eval 30.6%

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

      16. times-frac 30.6%

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

      17. neg-mul-1 30.6%

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

      18. distribute-frac-neg 30.6%

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

      19. fabs-neg 30.6%

          \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \color{blue}{\left|\frac{a}{3 \cdot b}\right|}\right) \]

      20. rem-square-sqrt 18.5%

          \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \left|\color{blue}{\sqrt{\frac{a}{3 \cdot b}} \cdot \sqrt{\frac{a}{3 \cdot b}}}\right|\right) \]

   7. Simplified 28.7%

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

   8. Taylor expanded in t around 0 27.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{a}{b} + 2 \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-49} \lor \neg \left(\frac{a}{3 \cdot b} \leq 10^{-124}\right):\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} + 0.3333333333333333 \cdot \frac{a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in z around 0 74.9%

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

4. Taylor expanded in y around 0 63.2%

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

   1. *-commutative 63.2%

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

6. Simplified 63.2%

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

7. Final simplification 63.2%

   \[\leadsto 2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \]
8. Add Preprocessing

Alternative 9: 50.9% accurate, 43.4× 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 70.5%

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

3. Simplified 69.9%

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

5. Taylor expanded in a around inf 51.3%

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

   1. *-commutative 51.3%

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

   2. associate-*l/ 51.3%

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

   3. associate-*r/ 51.3%

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

   4. clear-num 51.2%

      \[\leadsto a \cdot \color{blue}{\frac{1}{\frac{b}{-0.3333333333333333}}} \]

   5. un-div-inv 51.3%

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

   6. div-inv 51.4%

      \[\leadsto \frac{a}{\color{blue}{b \cdot \frac{1}{-0.3333333333333333}}} \]

   7. metadata-eval 51.4%

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

   8. metadata-eval 51.4%

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

   9. distribute-rgt-neg-in 51.4%

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

   10. *-commutative 51.4%

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

   11. distribute-lft-neg-in 51.4%

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

   12. metadata-eval 51.4%

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

7. Applied egg-rr 51.4%

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

8. Final simplification 51.4%

   \[\leadsto \frac{a}{b \cdot -3} \]
9. Add Preprocessing

Alternative 10: 50.8% accurate, 43.4× 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 70.5%

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

3. Simplified 69.9%

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

5. Taylor expanded in a around inf 51.3%

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

Developer Target 1: 74.3% accurate, 1.0× 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 2024152 
(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))))