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

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 5

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 86.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-91} \lor \neg \left(x \leq 2 \cdot 10^{-94}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.5e-91) (not (<= x 2e-94)))
   (fma 1.0 z (* (sin y) x))
   (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.5e-91) || !(x <= 2e-94)) {
		tmp = fma(1.0, z, (sin(y) * x));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.5e-91) || !(x <= 2e-94))
		tmp = fma(1.0, z, Float64(sin(y) * x));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.5e-91], N[Not[LessEqual[x, 2e-94]], $MachinePrecision]], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000051e-91 or 1.9999999999999999e-94 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \sin y \cdot x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 89.0%

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

   if -7.50000000000000051e-91 < x < 1.9999999999999999e-94

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-91} \lor \neg \left(x \leq 2 \cdot 10^{-94}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, x\right) \cdot y + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7) (not (<= y 0.5)))
   (* (cos y) z)
   (+
    (*
     (fma
      (* x (fma 0.008333333333333333 (* y y) -0.16666666666666666))
      (* y y)
      x)
     y)
    (*
     z
     (fma
      (fma
       (fma -0.001388888888888889 (* y y) 0.041666666666666664)
       (* y y)
       -0.5)
      (* y y)
      1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7) || !(y <= 0.5)) {
		tmp = cos(y) * z;
	} else {
		tmp = (fma((x * fma(0.008333333333333333, (y * y), -0.16666666666666666)), (y * y), x) * y) + (z * fma(fma(fma(-0.001388888888888889, (y * y), 0.041666666666666664), (y * y), -0.5), (y * y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.7) || !(y <= 0.5))
		tmp = Float64(cos(y) * z);
	else
		tmp = Float64(Float64(fma(Float64(x * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666)), Float64(y * y), x) * y) + Float64(z * fma(fma(fma(-0.001388888888888889, Float64(y * y), 0.041666666666666664), Float64(y * y), -0.5), Float64(y * y), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.7], N[Not[LessEqual[y, 0.5]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(N[(x * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * y), $MachinePrecision] + N[(z * N[(N[(N[(-0.001388888888888889 * N[(y * y), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.69999999999999996 or 0.5 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if -1.69999999999999996 < y < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot x + z \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot x + z \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + z \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot x + z \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)} \]

      4. sub-neg N/A

         \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right) \]

      8. +-commutative N/A

         \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right) \]

      15. lower-*.f64 99.5

          \[\leadsto y \cdot x + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right) \]

   8. Applied rewrites 99.5%

      \[\leadsto y \cdot x + z \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \cdot y} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), y \cdot y, 1\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \cdot y} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, x\right) \cdot y} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, x\right) \cdot y + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.5% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 51.2

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

5. Applied rewrites 51.2%

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

Alternative 5: 39.4% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 z))

C

double code(double x, double y, double z) {
	return 1.0 * 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 = 1.0d0 * z
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 * z;
}

Python

def code(x, y, z):
	return 1.0 * z

Julia

function code(x, y, z)
	return Float64(1.0 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 * z;
end

Wolfram

code[x_, y_, z_] := N[(1.0 * z), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(\frac{x \cdot \sin y}{z} + \left(\mathsf{neg}\left(-1 \cdot \cos y\right)\right)\right)} \cdot z \]

   3. mul-1-neg N/A

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

   4. remove-double-neg N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 90.5

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

7. Applied rewrites 90.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{x \cdot \sin y}{z} \cdot z \]
9. Step-by-step derivation

10. Applied rewrites 32.4%

    \[\leadsto \frac{\sin y \cdot x}{z} \cdot z \]

11. Taylor expanded in y around 0

    \[\leadsto 1 \cdot z \]
12. Step-by-step derivation

13. Applied rewrites 39.3%

    \[\leadsto 1 \cdot z \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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))))