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

Percentage Accurate: 99.8% → 99.8%

Time: 9.5s

Alternatives: 8

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

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

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: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-90} \lor \neg \left(x \leq -7 \cdot 10^{-137} \lor \neg \left(x \leq -1.9 \cdot 10^{-162}\right) \land x \leq 3000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e-90)
         (not (or (<= x -7e-137) (and (not (<= x -1.9e-162)) (<= x 3000.0)))))
   (* x (cos y))
   (* (sin y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e-90) || !((x <= -7e-137) || (!(x <= -1.9e-162) && (x <= 3000.0)))) {
		tmp = x * cos(y);
	} else {
		tmp = sin(y) * -z;
	}
	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 <= (-5d-90)) .or. (.not. (x <= (-7d-137)) .or. (.not. (x <= (-1.9d-162))) .and. (x <= 3000.0d0))) then
        tmp = x * cos(y)
    else
        tmp = sin(y) * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e-90) || !((x <= -7e-137) || (!(x <= -1.9e-162) && (x <= 3000.0)))) {
		tmp = x * Math.cos(y);
	} else {
		tmp = Math.sin(y) * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5e-90) or not ((x <= -7e-137) or (not (x <= -1.9e-162) and (x <= 3000.0))):
		tmp = x * math.cos(y)
	else:
		tmp = math.sin(y) * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e-90) || !((x <= -7e-137) || (!(x <= -1.9e-162) && (x <= 3000.0))))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(sin(y) * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5e-90) || ~(((x <= -7e-137) || (~((x <= -1.9e-162)) && (x <= 3000.0)))))
		tmp = x * cos(y);
	else
		tmp = sin(y) * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e-90], N[Not[Or[LessEqual[x, -7e-137], And[N[Not[LessEqual[x, -1.9e-162]], $MachinePrecision], LessEqual[x, 3000.0]]]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-90 or -7.0000000000000002e-137 < x < -1.90000000000000002e-162 or 3e3 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 85.8%

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

   if -5.00000000000000019e-90 < x < -7.0000000000000002e-137 or -1.90000000000000002e-162 < x < 3e3

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 72.7%

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

      1. neg-mul-1 72.7%

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

      2. distribute-lft-neg-in 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-90} \lor \neg \left(x \leq -7 \cdot 10^{-137} \lor \neg \left(x \leq -1.9 \cdot 10^{-162}\right) \land x \leq 3000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+21} \lor \neg \left(x \leq 2.6 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.4e+21) (not (<= x 2.6e+113)))
   (* x (cos y))
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e+21) || !(x <= 2.6e+113)) {
		tmp = x * cos(y);
	} else {
		tmp = x - (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) :: tmp
    if ((x <= (-3.4d+21)) .or. (.not. (x <= 2.6d+113))) then
        tmp = x * cos(y)
    else
        tmp = x - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e+21) || !(x <= 2.6e+113)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.4e+21) or not (x <= 2.6e+113):
		tmp = x * math.cos(y)
	else:
		tmp = x - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.4e+21) || !(x <= 2.6e+113))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.4e+21) || ~((x <= 2.6e+113)))
		tmp = x * cos(y);
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.4e+21], N[Not[LessEqual[x, 2.6e+113]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e21 or 2.5999999999999999e113 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 90.1%

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

   if -3.4e21 < x < 2.5999999999999999e113

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 88.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+21} \lor \neg \left(x \leq 2.6 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0064 \lor \neg \left(y \leq 0.006\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0064) (not (<= y 0.006)))
   (* x (cos y))
   (+ x (* y (- (* y (* y (* z 0.16666666666666666))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0064) || !(y <= 0.006)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * ((y * (y * (z * 0.16666666666666666))) - z));
	}
	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 <= (-0.0064d0)) .or. (.not. (y <= 0.006d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * ((y * (y * (z * 0.16666666666666666d0))) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0064) || !(y <= 0.006)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((y * (y * (z * 0.16666666666666666))) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0064) or not (y <= 0.006):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((y * (y * (z * 0.16666666666666666))) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0064) || !(y <= 0.006))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(y * Float64(z * 0.16666666666666666))) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0064) || ~((y <= 0.006)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((y * (y * (z * 0.16666666666666666))) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0064], N[Not[LessEqual[y, 0.006]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00640000000000000031 or 0.0060000000000000001 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 49.0%

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

   if -0.00640000000000000031 < y < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot \left(y \cdot z\right)\right) - z\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto x + y \cdot \left(y \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot z\right)\right)} - z\right) \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   6. Simplified 100.0%

      \[\leadsto x + y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(z \cdot 0.16666666666666666\right)\right)} - z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0064 \lor \neg \left(y \leq 0.006\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(y \cdot \left(z \cdot 0.16666666666666666\right)\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.3% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+96} \lor \neg \left(z \leq 4.8 \cdot 10^{+129}\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e+96) (not (<= z 4.8e+129))) (* z (- y)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e+96) || !(z <= 4.8e+129)) {
		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 ((z <= (-1.75d+96)) .or. (.not. (z <= 4.8d+129))) 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 ((z <= -1.75e+96) || !(z <= 4.8e+129)) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e+96) or not (z <= 4.8e+129):
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e+96) || !(z <= 4.8e+129))
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e+96) || ~((z <= 4.8e+129)))
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e+96], N[Not[LessEqual[z, 4.8e+129]], $MachinePrecision]], N[(z * (-y)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.7499999999999999e96 or 4.7999999999999997e129 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 75.3%

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

      1. neg-mul-1 75.3%

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

      2. distribute-lft-neg-in 75.3%

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

   5. Simplified 75.3%

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

   6. Taylor expanded in y around 0 30.8%

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

      1. associate-*r* 30.8%

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

      2. mul-1-neg 30.8%

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

   8. Simplified 30.8%

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

   if -1.7499999999999999e96 < z < 4.7999999999999997e129

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/l* 81.9%

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

      3. fma-neg 81.9%

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

   5. Simplified 81.9%

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

   6. Taylor expanded in y around 0 43.7%

      \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-0.5 \cdot \frac{x}{z} + 0.16666666666666666 \cdot y\right) - 1\right) + \frac{x}{z}\right)} \]

   7. Taylor expanded in y around 0 52.3%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+96} \lor \neg \left(z \leq 4.8 \cdot 10^{+129}\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.1% 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 54.4%

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

   1. mul-1-neg 54.4%

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

   2. unsub-neg 54.4%

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

5. Simplified 54.4%

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

6. Final simplification 54.4%

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

Alternative 8: 39.2% 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 z around inf 86.7%

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

   1. *-commutative 86.7%

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

   2. associate-/l* 86.7%

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

   3. fma-neg 86.7%

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

5. Simplified 86.7%

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

6. Taylor expanded in y around 0 44.2%

   \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-0.5 \cdot \frac{x}{z} + 0.16666666666666666 \cdot y\right) - 1\right) + \frac{x}{z}\right)} \]

7. Taylor expanded in y around 0 42.8%

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

8. Final simplification 42.8%

   \[\leadsto x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(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))))