Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.1112659826902206e+266
  2. y = 2.497032862420723e+141
  3. z = 3.1137565307214543e+302

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-define 99.8%

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

   7. sin-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0003:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-6}:\\ \;\;\;\;x - z \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+32} \lor \neg \left(y \leq 2.8 \cdot 10^{+137}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0003)
     t_0
     (if (<= y 9.8e-6)
       (- x (* z y))
       (if (or (<= y 4.2e+32) (not (<= y 2.8e+137))) t_0 (* z (- (sin y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.0003) {
		tmp = t_0;
	} else if (y <= 9.8e-6) {
		tmp = x - (z * y);
	} else if ((y <= 4.2e+32) || !(y <= 2.8e+137)) {
		tmp = t_0;
	} else {
		tmp = z * -sin(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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (y <= (-0.0003d0)) then
        tmp = t_0
    else if (y <= 9.8d-6) then
        tmp = x - (z * y)
    else if ((y <= 4.2d+32) .or. (.not. (y <= 2.8d+137))) then
        tmp = t_0
    else
        tmp = z * -sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (y <= -0.0003) {
		tmp = t_0;
	} else if (y <= 9.8e-6) {
		tmp = x - (z * y);
	} else if ((y <= 4.2e+32) || !(y <= 2.8e+137)) {
		tmp = t_0;
	} else {
		tmp = z * -Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -0.0003:
		tmp = t_0
	elif y <= 9.8e-6:
		tmp = x - (z * y)
	elif (y <= 4.2e+32) or not (y <= 2.8e+137):
		tmp = t_0
	else:
		tmp = z * -math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.0003)
		tmp = t_0;
	elseif (y <= 9.8e-6)
		tmp = Float64(x - Float64(z * y));
	elseif ((y <= 4.2e+32) || !(y <= 2.8e+137))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -0.0003)
		tmp = t_0;
	elseif (y <= 9.8e-6)
		tmp = x - (z * y);
	elseif ((y <= 4.2e+32) || ~((y <= 2.8e+137)))
		tmp = t_0;
	else
		tmp = z * -sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0003], t$95$0, If[LessEqual[y, 9.8e-6], N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 4.2e+32], N[Not[LessEqual[y, 2.8e+137]], $MachinePrecision]], t$95$0, N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.99999999999999974e-4 or 9.79999999999999934e-6 < y < 4.2000000000000001e32 or 2.80000000000000001e137 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 60.7%

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

   if -2.99999999999999974e-4 < y < 9.79999999999999934e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.7%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

   8. Simplified 99.7%

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

   if 4.2000000000000001e32 < y < 2.80000000000000001e137

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 68.7%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 68.7%

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

      2. *-commutative 68.7%

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

      3. distribute-rgt-neg-in 68.7%

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

   5. Simplified 68.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0003:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-6}:\\ \;\;\;\;x - z \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+32} \lor \neg \left(y \leq 2.8 \cdot 10^{+137}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.00033:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00023:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+32} \lor \neg \left(y \leq 8.6 \cdot 10^{+137}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.00033)
     t_0
     (if (<= y 0.00023)
       (fma (- y) z x)
       (if (or (<= y 2.1e+32) (not (<= y 8.6e+137))) t_0 (* z (- (sin y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.00033) {
		tmp = t_0;
	} else if (y <= 0.00023) {
		tmp = fma(-y, z, x);
	} else if ((y <= 2.1e+32) || !(y <= 8.6e+137)) {
		tmp = t_0;
	} else {
		tmp = z * -sin(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.00033)
		tmp = t_0;
	elseif (y <= 0.00023)
		tmp = fma(Float64(-y), z, x);
	elseif ((y <= 2.1e+32) || !(y <= 8.6e+137))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-sin(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00033], t$95$0, If[LessEqual[y, 0.00023], N[((-y) * z + x), $MachinePrecision], If[Or[LessEqual[y, 2.1e+32], N[Not[LessEqual[y, 8.6e+137]], $MachinePrecision]], t$95$0, N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3e-4 or 2.3000000000000001e-4 < y < 2.1000000000000001e32 or 8.59999999999999929e137 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 60.7%

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

   if -3.3e-4 < y < 2.3000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.7%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + x} \]

      2. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot z} + x \]

      3. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, z, x\right)} \]

      4. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   if 2.1000000000000001e32 < y < 8.59999999999999929e137

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 68.7%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 68.7%

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

      2. *-commutative 68.7%

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

      3. distribute-rgt-neg-in 68.7%

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

   5. Simplified 68.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00033:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.00023:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+32} \lor \neg \left(y \leq 8.6 \cdot 10^{+137}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00022:\\ \;\;\;\;x - \left(-0.16666666666666666 \cdot \left(z \cdot {y}^{3}\right) + z \cdot y\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+32} \lor \neg \left(y \leq 7.2 \cdot 10^{+137}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0105)
     t_0
     (if (<= y 0.00022)
       (- x (+ (* -0.16666666666666666 (* z (pow y 3.0))) (* z y)))
       (if (or (<= y 2.1e+32) (not (<= y 7.2e+137))) t_0 (* z (- (sin y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.0105) {
		tmp = t_0;
	} else if (y <= 0.00022) {
		tmp = x - ((-0.16666666666666666 * (z * pow(y, 3.0))) + (z * y));
	} else if ((y <= 2.1e+32) || !(y <= 7.2e+137)) {
		tmp = t_0;
	} else {
		tmp = z * -sin(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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (y <= (-0.0105d0)) then
        tmp = t_0
    else if (y <= 0.00022d0) then
        tmp = x - (((-0.16666666666666666d0) * (z * (y ** 3.0d0))) + (z * y))
    else if ((y <= 2.1d+32) .or. (.not. (y <= 7.2d+137))) then
        tmp = t_0
    else
        tmp = z * -sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (y <= -0.0105) {
		tmp = t_0;
	} else if (y <= 0.00022) {
		tmp = x - ((-0.16666666666666666 * (z * Math.pow(y, 3.0))) + (z * y));
	} else if ((y <= 2.1e+32) || !(y <= 7.2e+137)) {
		tmp = t_0;
	} else {
		tmp = z * -Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -0.0105:
		tmp = t_0
	elif y <= 0.00022:
		tmp = x - ((-0.16666666666666666 * (z * math.pow(y, 3.0))) + (z * y))
	elif (y <= 2.1e+32) or not (y <= 7.2e+137):
		tmp = t_0
	else:
		tmp = z * -math.sin(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.0105)
		tmp = t_0;
	elseif (y <= 0.00022)
		tmp = Float64(x - Float64(Float64(-0.16666666666666666 * Float64(z * (y ^ 3.0))) + Float64(z * y)));
	elseif ((y <= 2.1e+32) || !(y <= 7.2e+137))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -0.0105)
		tmp = t_0;
	elseif (y <= 0.00022)
		tmp = x - ((-0.16666666666666666 * (z * (y ^ 3.0))) + (z * y));
	elseif ((y <= 2.1e+32) || ~((y <= 7.2e+137)))
		tmp = t_0;
	else
		tmp = z * -sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0105], t$95$0, If[LessEqual[y, 0.00022], N[(x - N[(N[(-0.16666666666666666 * N[(z * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 2.1e+32], N[Not[LessEqual[y, 7.2e+137]], $MachinePrecision]], t$95$0, N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0105000000000000007 or 2.20000000000000008e-4 < y < 2.1000000000000001e32 or 7.1999999999999999e137 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 60.7%

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

   if -0.0105000000000000007 < y < 2.20000000000000008e-4

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

         \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{x} - z \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right) \]

   6. Taylor expanded in y around 0 100.0%

      \[\leadsto x - \color{blue}{\left(-0.16666666666666666 \cdot \left({y}^{3} \cdot z\right) + y \cdot z\right)} \]

   if 2.1000000000000001e32 < y < 7.1999999999999999e137

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 68.7%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 68.7%

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

      2. *-commutative 68.7%

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

      3. distribute-rgt-neg-in 68.7%

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

   5. Simplified 68.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0105:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.00022:\\ \;\;\;\;x - \left(-0.16666666666666666 \cdot \left(z \cdot {y}^{3}\right) + z \cdot y\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+32} \lor \neg \left(y \leq 7.2 \cdot 10^{+137}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00115 \lor \neg \left(y \leq 0.0001\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00115) (not (<= y 0.0001))) (* x (cos y)) (- x (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00115) || !(y <= 0.0001)) {
		tmp = x * cos(y);
	} else {
		tmp = x - (z * 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 ((y <= (-0.00115d0)) .or. (.not. (y <= 0.0001d0))) then
        tmp = x * cos(y)
    else
        tmp = x - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00115) || !(y <= 0.0001)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00115) or not (y <= 0.0001):
		tmp = x * math.cos(y)
	else:
		tmp = x - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00115) || !(y <= 0.0001))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00115) || ~((y <= 0.0001)))
		tmp = x * cos(y);
	else
		tmp = x - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00115], N[Not[LessEqual[y, 0.0001]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00115 or 1.00000000000000005e-4 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 55.0%

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

   if -0.00115 < y < 1.00000000000000005e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.7%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.7%

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

      2. sub-neg 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00115 \lor \neg \left(y \leq 0.0001\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.8% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-181}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e-90) x (if (<= x 3.5e-181) (* z (- y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e-90) {
		tmp = x;
	} else if (x <= 3.5e-181) {
		tmp = z * -y;
	} 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 <= (-1.2d-90)) then
        tmp = x
    else if (x <= 3.5d-181) then
        tmp = z * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e-90) {
		tmp = x;
	} else if (x <= 3.5e-181) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.2e-90:
		tmp = x
	elif x <= 3.5e-181:
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e-90)
		tmp = x;
	elseif (x <= 3.5e-181)
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.2e-90)
		tmp = x;
	elseif (x <= 3.5e-181)
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e-90], x, If[LessEqual[x, 3.5e-181], N[(z * (-y)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e-90 or 3.49999999999999996e-181 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 53.9%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 53.9%

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

   5. Simplified 53.9%

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

   6. Taylor expanded in x around inf 47.6%

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

   if -1.2000000000000001e-90 < x < 3.49999999999999996e-181

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 44.1%

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 44.1%

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

   5. Simplified 44.1%

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

   6. Taylor expanded in x around 0 31.9%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 31.9%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

      2. neg-mul-1 31.9%

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

   8. Simplified 31.9%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-181}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 41.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x - (z * 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 - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - N[(z * y), $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 51.1%

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 51.1%

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

5. Simplified 51.1%

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

6. Taylor expanded in x around 0 51.1%

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 51.1%

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

   2. sub-neg 51.1%

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

8. Simplified 51.1%

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

9. Final simplification 51.1%

   \[\leadsto x - z \cdot y \]
10. Add Preprocessing

Alternative 9: 39.1% 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 51.1%

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 51.1%

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

5. Simplified 51.1%

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

6. Taylor expanded in x around inf 38.9%

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

7. Final simplification 38.9%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024045 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))