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

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * 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 = (x * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * 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 = (x * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * 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 = (x * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]

2. Final simplification 99.8%

   \[\leadsto x \cdot \sin y + z \cdot \cos y \]

Alternative 2: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+91} \lor \neg \left(z \leq 5.2 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.6e+91) (not (<= z 5.2e-21)))
   (* z (cos y))
   (+ (* x (sin y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.6e+91) || !(z <= 5.2e-21)) {
		tmp = z * cos(y);
	} else {
		tmp = (x * 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 ((z <= (-8.6d+91)) .or. (.not. (z <= 5.2d-21))) then
        tmp = z * cos(y)
    else
        tmp = (x * sin(y)) + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.6e+91) || !(z <= 5.2e-21)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = (x * Math.sin(y)) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.6e+91) or not (z <= 5.2e-21):
		tmp = z * math.cos(y)
	else:
		tmp = (x * math.sin(y)) + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.6e+91) || !(z <= 5.2e-21))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(x * sin(y)) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.6e+91) || ~((z <= 5.2e-21)))
		tmp = z * cos(y);
	else
		tmp = (x * sin(y)) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.6e+91], N[Not[LessEqual[z, 5.2e-21]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.6000000000000001e91 or 5.20000000000000035e-21 < z

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in x around 0 91.7%

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

   if -8.6000000000000001e91 < z < 5.20000000000000035e-21

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 87.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+91} \lor \neg \left(z \leq 5.2 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z\\ \end{array} \]

Alternative 3: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \lor \neg \left(y \leq 0.0095\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.7) (not (<= y 0.0095)))
   (* x (sin y))
   (+ z (* y (+ x (* y (* z -0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7) || !(y <= 0.0095)) {
		tmp = x * sin(y);
	} else {
		tmp = z + (y * (x + (y * (z * -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 <= (-3.7d0)) .or. (.not. (y <= 0.0095d0))) then
        tmp = x * sin(y)
    else
        tmp = z + (y * (x + (y * (z * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7) || !(y <= 0.0095)) {
		tmp = x * Math.sin(y);
	} else {
		tmp = z + (y * (x + (y * (z * -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.7) or not (y <= 0.0095):
		tmp = x * math.sin(y)
	else:
		tmp = z + (y * (x + (y * (z * -0.5))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.7) || !(y <= 0.0095))
		tmp = Float64(x * sin(y));
	else
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(z * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.7) || ~((y <= 0.0095)))
		tmp = x * sin(y);
	else
		tmp = z + (y * (x + (y * (z * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.7], N[Not[LessEqual[y, 0.0095]], $MachinePrecision]], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + N[(y * N[(x + N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7000000000000002 or 0.00949999999999999976 < y

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in x around inf 48.3%

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

   if -3.7000000000000002 < y < 0.00949999999999999976

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. add-cube-cbrt 100.0%

         \[\leadsto z \cdot \color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) \cdot \sqrt[3]{\cos y}\right)} + x \cdot \sin y \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}} + x \cdot \sin y \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right), \sqrt[3]{\cos y}, x \cdot \sin y\right)} \]

      5. pow2 100.0%

         \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{2}}, \sqrt[3]{\cos y}, x \cdot \sin y\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot {\left(\sqrt[3]{\cos y}\right)}^{2}, \sqrt[3]{\cos y}, x \cdot \sin y\right)} \]

   4. Taylor expanded in y around 0 98.8%

      \[\leadsto \color{blue}{z + \left(x \cdot y + {y}^{2} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto z + \left(\color{blue}{y \cdot x} + {y}^{2} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right) \]

      2. unpow2 98.8%

         \[\leadsto z + \left(y \cdot x + \color{blue}{\left(y \cdot y\right)} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right) \]

      3. associate-*l* 98.8%

         \[\leadsto z + \left(y \cdot x + \color{blue}{y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)}\right) \]

      4. distribute-lft-out 98.8%

         \[\leadsto z + \color{blue}{y \cdot \left(x + y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)} \]

      5. distribute-rgt-out 98.8%

         \[\leadsto z + y \cdot \left(x + y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 + -0.16666666666666666\right)\right)}\right) \]

      6. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \lor \neg \left(y \leq 0.0095\right):\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 54000000000:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.7)
   (* x (sin y))
   (if (<= y 54000000000.0) (+ z (* y (+ x (* y (* z -0.5))))) (* z (cos y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7) {
		tmp = x * sin(y);
	} else if (y <= 54000000000.0) {
		tmp = z + (y * (x + (y * (z * -0.5))));
	} else {
		tmp = z * 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 (y <= (-3.7d0)) then
        tmp = x * sin(y)
    else if (y <= 54000000000.0d0) then
        tmp = z + (y * (x + (y * (z * (-0.5d0)))))
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7) {
		tmp = x * Math.sin(y);
	} else if (y <= 54000000000.0) {
		tmp = z + (y * (x + (y * (z * -0.5))));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.7:
		tmp = x * math.sin(y)
	elif y <= 54000000000.0:
		tmp = z + (y * (x + (y * (z * -0.5))))
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.7)
		tmp = Float64(x * sin(y));
	elseif (y <= 54000000000.0)
		tmp = Float64(z + Float64(y * Float64(x + Float64(y * Float64(z * -0.5)))));
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.7)
		tmp = x * sin(y);
	elseif (y <= 54000000000.0)
		tmp = z + (y * (x + (y * (z * -0.5))));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.7], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 54000000000.0], N[(z + N[(y * N[(x + N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7000000000000002

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in x around inf 52.3%

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

   if -3.7000000000000002 < y < 5.4e10

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. add-cube-cbrt 100.0%

         \[\leadsto z \cdot \color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) \cdot \sqrt[3]{\cos y}\right)} + x \cdot \sin y \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}} + x \cdot \sin y \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right), \sqrt[3]{\cos y}, x \cdot \sin y\right)} \]

      5. pow2 100.0%

         \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{2}}, \sqrt[3]{\cos y}, x \cdot \sin y\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot {\left(\sqrt[3]{\cos y}\right)}^{2}, \sqrt[3]{\cos y}, x \cdot \sin y\right)} \]

   4. Taylor expanded in y around 0 97.6%

      \[\leadsto \color{blue}{z + \left(x \cdot y + {y}^{2} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto z + \left(\color{blue}{y \cdot x} + {y}^{2} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right) \]

      2. unpow2 97.6%

         \[\leadsto z + \left(y \cdot x + \color{blue}{\left(y \cdot y\right)} \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right) \]

      3. associate-*l* 97.6%

         \[\leadsto z + \left(y \cdot x + \color{blue}{y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)}\right) \]

      4. distribute-lft-out 97.6%

         \[\leadsto z + \color{blue}{y \cdot \left(x + y \cdot \left(-0.3333333333333333 \cdot z + -0.16666666666666666 \cdot z\right)\right)} \]

      5. distribute-rgt-out 97.6%

         \[\leadsto z + y \cdot \left(x + y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 + -0.16666666666666666\right)\right)}\right) \]

      6. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   if 5.4e10 < y

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in x around 0 59.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 54000000000:\\ \;\;\;\;z + y \cdot \left(x + y \cdot \left(z \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 5: 52.9% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]

2. Taylor expanded in y around 0 57.5%

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

   1. +-commutative 57.5%

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

4. Simplified 57.5%

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

5. Final simplification 57.5%

   \[\leadsto z + x \cdot y \]

Alternative 6: 39.4% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. add-cube-cbrt 99.4%

      \[\leadsto z \cdot \color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) \cdot \sqrt[3]{\cos y}\right)} + x \cdot \sin y \]

   3. associate-*r* 99.4%

      \[\leadsto \color{blue}{\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}} + x \cdot \sin y \]

   4. fma-def 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right), \sqrt[3]{\cos y}, x \cdot \sin y\right)} \]

   5. pow2 99.4%

      \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{2}}, \sqrt[3]{\cos y}, x \cdot \sin y\right) \]

3. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot {\left(\sqrt[3]{\cos y}\right)}^{2}, \sqrt[3]{\cos y}, x \cdot \sin y\right)} \]

4. Taylor expanded in y around 0 44.1%

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

5. Final simplification 44.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))