Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \cos y + z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) + (z * sin(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) + (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.cos(y)) + (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x * math.cos(y)) + (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) + Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) + (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos y + z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) + (z * sin(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) + (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.cos(y)) + (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x * math.cos(y)) + (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) + Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) + (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \cos y, z \cdot \sin y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (cos y) (* z (sin y))))

C

double code(double x, double y, double z) {
	return fma(x, cos(y), (z * sin(y)));
}

Julia

function code(x, y, z)
	return fma(x, cos(y), Float64(z * sin(y)))
end

Wolfram

code[x_, y_, z_] := N[(x * N[Cos[y], $MachinePrecision] + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
2. Step-by-step derivation

   1. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x, \cos y, z \cdot \sin y\right) \]
6. Add Preprocessing

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.52e-149)
   (fma (sin y) z x)
   (if (<= z 4.5e-51) (* x (cos y)) (+ x (* z (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.52e-149) {
		tmp = fma(sin(y), z, x);
	} else if (z <= 4.5e-51) {
		tmp = x * cos(y);
	} else {
		tmp = x + (z * sin(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.52e-149)
		tmp = fma(sin(y), z, x);
	elseif (z <= 4.5e-51)
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(z * sin(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.52e-149], N[(N[Sin[y], $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 4.5e-51], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5199999999999999e-149

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 91.5%

      \[\leadsto \color{blue}{x} + z \cdot \sin y \]
   4. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto \color{blue}{z \cdot \sin y + x} \]

      2. *-commutative 91.5%

         \[\leadsto \color{blue}{\sin y \cdot z} + x \]

      3. fma-def 91.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, z, x\right)} \]

   5. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, z, x\right)} \]

   if -1.5199999999999999e-149 < z < 4.49999999999999974e-51

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.3%

      \[\leadsto \color{blue}{x \cdot \cos y} \]

   if 4.49999999999999974e-51 < z

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 89.8%

      \[\leadsto \color{blue}{x} + z \cdot \sin y \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ z \cdot \sin y + x \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* z (sin y)) (* x (cos y))))

C

double code(double x, double y, double z) {
	return (z * sin(y)) + (x * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z * sin(y)) + (x * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (z * Math.sin(y)) + (x * Math.cos(y));
}

Python

def code(x, y, z):
	return (z * math.sin(y)) + (x * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(z * sin(y)) + Float64(x * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (z * sin(y)) + (x * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto z \cdot \sin y + x \cdot \cos y \]
4. Add Preprocessing

Alternative 4: 74.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -4 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+91} \lor \neg \left(y \leq 1.7 \cdot 10^{+204}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= y -4e+148)
     t_0
     (if (<= y -1.42e+20)
       t_1
       (if (<= y 6.5e-18)
         (+ x (* y (+ z (* y (* x -0.5)))))
         (if (or (<= y 1.36e+91) (not (<= y 1.7e+204))) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -4e+148) {
		tmp = t_0;
	} else if (y <= -1.42e+20) {
		tmp = t_1;
	} else if (y <= 6.5e-18) {
		tmp = x + (y * (z + (y * (x * -0.5))));
	} else if ((y <= 1.36e+91) || !(y <= 1.7e+204)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = x * cos(y)
    if (y <= (-4d+148)) then
        tmp = t_0
    else if (y <= (-1.42d+20)) then
        tmp = t_1
    else if (y <= 6.5d-18) then
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    else if ((y <= 1.36d+91) .or. (.not. (y <= 1.7d+204))) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -4e+148) {
		tmp = t_0;
	} else if (y <= -1.42e+20) {
		tmp = t_1;
	} else if (y <= 6.5e-18) {
		tmp = x + (y * (z + (y * (x * -0.5))));
	} else if ((y <= 1.36e+91) || !(y <= 1.7e+204)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -4e+148:
		tmp = t_0
	elif y <= -1.42e+20:
		tmp = t_1
	elif y <= 6.5e-18:
		tmp = x + (y * (z + (y * (x * -0.5))))
	elif (y <= 1.36e+91) or not (y <= 1.7e+204):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -4e+148)
		tmp = t_0;
	elseif (y <= -1.42e+20)
		tmp = t_1;
	elseif (y <= 6.5e-18)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	elseif ((y <= 1.36e+91) || !(y <= 1.7e+204))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -4e+148)
		tmp = t_0;
	elseif (y <= -1.42e+20)
		tmp = t_1;
	elseif (y <= 6.5e-18)
		tmp = x + (y * (z + (y * (x * -0.5))));
	elseif ((y <= 1.36e+91) || ~((y <= 1.7e+204)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+148], t$95$0, If[LessEqual[y, -1.42e+20], t$95$1, If[LessEqual[y, 6.5e-18], N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.36e+91], N[Not[LessEqual[y, 1.7e+204]], $MachinePrecision]], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0000000000000002e148 or 6.50000000000000008e-18 < y < 1.36000000000000007e91 or 1.70000000000000005e204 < y

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.3%

      \[\leadsto \color{blue}{z \cdot \sin y} \]

   if -4.0000000000000002e148 < y < -1.42e20 or 1.36000000000000007e91 < y < 1.70000000000000005e204

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 70.2%

      \[\leadsto \color{blue}{x \cdot \cos y} \]

   if -1.42e20 < y < 6.50000000000000008e-18

   1. Initial program 100.0%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.6%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot \left(x \cdot {y}^{2}\right) + y \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.6%

         \[\leadsto x + \color{blue}{\left(y \cdot z + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right)} \]

      2. *-commutative 98.6%

         \[\leadsto x + \left(\color{blue}{z \cdot y} + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right) \]

      3. associate-*r* 98.6%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(-0.5 \cdot x\right) \cdot {y}^{2}}\right) \]

      4. unpow2 98.6%

         \[\leadsto x + \left(z \cdot y + \left(-0.5 \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      5. associate-*r* 98.6%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(\left(-0.5 \cdot x\right) \cdot y\right) \cdot y}\right) \]

      6. distribute-rgt-out 98.6%

         \[\leadsto x + \color{blue}{y \cdot \left(z + \left(-0.5 \cdot x\right) \cdot y\right)} \]

      7. *-commutative 98.6%

         \[\leadsto x + y \cdot \left(z + \color{blue}{\left(x \cdot -0.5\right)} \cdot y\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{x + y \cdot \left(z + \left(x \cdot -0.5\right) \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+148}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+91} \lor \neg \left(y \leq 1.7 \cdot 10^{+204}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-149} \lor \neg \left(z \leq 8.5 \cdot 10^{-53}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.9e-149) (not (<= z 8.5e-53)))
   (+ x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e-149) || !(z <= 8.5e-53)) {
		tmp = x + (z * sin(y));
	} else {
		tmp = x * cos(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.9d-149)) .or. (.not. (z <= 8.5d-53))) then
        tmp = x + (z * sin(y))
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e-149) || !(z <= 8.5e-53)) {
		tmp = x + (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.9e-149) or not (z <= 8.5e-53):
		tmp = x + (z * math.sin(y))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.9e-149) || !(z <= 8.5e-53))
		tmp = Float64(x + Float64(z * sin(y)));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.9e-149) || ~((z <= 8.5e-53)))
		tmp = x + (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.9e-149], N[Not[LessEqual[z, 8.5e-53]], $MachinePrecision]], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9000000000000002e-149 or 8.50000000000000044e-53 < z

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 90.8%

      \[\leadsto \color{blue}{x} + z \cdot \sin y \]

   if -3.9000000000000002e-149 < z < 8.50000000000000044e-53

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 88.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-149} \lor \neg \left(z \leq 8.5 \cdot 10^{-53}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+20} \lor \neg \left(y \leq 1050000000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.42e+20) (not (<= y 1050000000.0)))
   (* x (cos y))
   (+ x (* y (+ z (* y (* x -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.42e+20) || !(y <= 1050000000.0)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.42d+20)) .or. (.not. (y <= 1050000000.0d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (y * (x * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.42e+20) || !(y <= 1050000000.0)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (y * (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.42e+20) or not (y <= 1050000000.0):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (y * (x * -0.5))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.42e+20) || !(y <= 1050000000.0))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(x * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.42e+20) || ~((y <= 1050000000.0)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (y * (x * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.42e+20], N[Not[LessEqual[y, 1050000000.0]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.42e20 or 1.05e9 < y

   1. Initial program 99.6%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 50.6%

      \[\leadsto \color{blue}{x \cdot \cos y} \]

   if -1.42e20 < y < 1.05e9

   1. Initial program 100.0%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 97.3%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot \left(x \cdot {y}^{2}\right) + y \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative 97.3%

         \[\leadsto x + \color{blue}{\left(y \cdot z + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right)} \]

      2. *-commutative 97.3%

         \[\leadsto x + \left(\color{blue}{z \cdot y} + -0.5 \cdot \left(x \cdot {y}^{2}\right)\right) \]

      3. associate-*r* 97.3%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(-0.5 \cdot x\right) \cdot {y}^{2}}\right) \]

      4. unpow2 97.3%

         \[\leadsto x + \left(z \cdot y + \left(-0.5 \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      5. associate-*r* 97.3%

         \[\leadsto x + \left(z \cdot y + \color{blue}{\left(\left(-0.5 \cdot x\right) \cdot y\right) \cdot y}\right) \]

      6. distribute-rgt-out 97.3%

         \[\leadsto x + \color{blue}{y \cdot \left(z + \left(-0.5 \cdot x\right) \cdot y\right)} \]

      7. *-commutative 97.3%

         \[\leadsto x + y \cdot \left(z + \color{blue}{\left(x \cdot -0.5\right)} \cdot y\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x + y \cdot \left(z + \left(x \cdot -0.5\right) \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+20} \lor \neg \left(y \leq 1050000000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-18}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-119) x (if (<= x 1e-18) (* y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-119) {
		tmp = x;
	} else if (x <= 1e-18) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.45d-119)) then
        tmp = x
    else if (x <= 1d-18) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-119) {
		tmp = x;
	} else if (x <= 1e-18) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-119:
		tmp = x
	elif x <= 1e-18:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-119)
		tmp = x;
	elseif (x <= 1e-18)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-119)
		tmp = x;
	elseif (x <= 1e-18)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-119], x, If[LessEqual[x, 1e-18], N[(y * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-119 or 1.0000000000000001e-18 < x

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 71.6%

      \[\leadsto \color{blue}{x} + z \cdot \sin y \]

   4. Taylor expanded in x around inf 52.5%

      \[\leadsto \color{blue}{x} \]

   if -1.45e-119 < x < 1.0000000000000001e-18

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 56.2%

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative 56.2%

         \[\leadsto \color{blue}{y \cdot z + x} \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{y \cdot z + x} \]

   6. Taylor expanded in y around inf 39.2%

      \[\leadsto \color{blue}{y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-18}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ x + y \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* y z)))

C

double code(double x, double y, double z) {
	return x + (y * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + (y * z);
}

Python

def code(x, y, z):
	return x + (y * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(y * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
2. Add Preprocessing

3. Taylor expanded in y around 0 58.1%

   \[\leadsto \color{blue}{x + y \cdot z} \]
4. Step-by-step derivation

   1. +-commutative 58.1%

      \[\leadsto \color{blue}{y \cdot z + x} \]

5. Simplified 58.1%

   \[\leadsto \color{blue}{y \cdot z + x} \]

6. Final simplification 58.1%

   \[\leadsto x + y \cdot z \]
7. Add Preprocessing

Alternative 9: 39.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
2. Add Preprocessing

3. Taylor expanded in y around 0 80.3%

   \[\leadsto \color{blue}{x} + z \cdot \sin y \]

4. Taylor expanded in x around inf 40.1%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 40.1%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))