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

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * x + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 73.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.84:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -1.65e+171)
     t_0
     (if (<= y -0.84)
       (* z (sin y))
       (if (<= y 8e-20) (fma (fma (* x y) -0.5 z) y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -1.65e+171) {
		tmp = t_0;
	} else if (y <= -0.84) {
		tmp = z * sin(y);
	} else if (y <= 8e-20) {
		tmp = fma(fma((x * y), -0.5, z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.65e+171)
		tmp = t_0;
	elseif (y <= -0.84)
		tmp = Float64(z * sin(y));
	elseif (y <= 8e-20)
		tmp = fma(fma(Float64(x * y), -0.5, z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.65e+171], t$95$0, If[LessEqual[y, -0.84], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-20], N[(N[(N[(x * y), $MachinePrecision] * -0.5 + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.64999999999999996e171 or 7.99999999999999956e-20 < y

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. sqrt-prod N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 39.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 39.7%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \cos y}\right)\right) \cdot x \]

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 64.4

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

   9. Applied rewrites 64.4%

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

   if -1.64999999999999996e171 < y < -0.839999999999999969

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if -0.839999999999999969 < y < 7.99999999999999956e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.84:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) z (* 1.0 x))))
   (if (<= z -7.8e-143) t_0 (if (<= z 1.52e-99) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), z, (1.0 * x));
	double tmp;
	if (z <= -7.8e-143) {
		tmp = t_0;
	} else if (z <= 1.52e-99) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), z, Float64(1.0 * x))
	tmp = 0.0
	if (z <= -7.8e-143)
		tmp = t_0;
	elseif (z <= 1.52e-99)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e-143], t$95$0, If[LessEqual[z, 1.52e-99], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.80000000000000007e-143 or 1.51999999999999999e-99 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 85.6%

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

   if -7.80000000000000007e-143 < z < 1.51999999999999999e-99

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. sqrt-prod N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 38.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 38.2%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \cos y}\right)\right) \cdot x \]

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 94.7

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

   9. Applied rewrites 94.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -0.84)
     t_0
     (if (<= y 8.6e+24) (fma (fma (* x y) -0.5 z) y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -0.84) {
		tmp = t_0;
	} else if (y <= 8.6e+24) {
		tmp = fma(fma((x * y), -0.5, z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -0.84)
		tmp = t_0;
	elseif (y <= 8.6e+24)
		tmp = fma(fma(Float64(x * y), -0.5, z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.84], t$95$0, If[LessEqual[y, 8.6e+24], N[(N[(N[(x * y), $MachinePrecision] * -0.5 + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.839999999999999969 or 8.59999999999999975e24 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 45.7

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

   5. Applied rewrites 45.7%

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

   if -0.839999999999999969 < y < 8.59999999999999975e24

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.1% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 53.9

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

5. Applied rewrites 53.9%

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

Alternative 7: 16.9% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 53.9

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

5. Applied rewrites 53.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 17.4%

   \[\leadsto z \cdot \color{blue}{y} \]
9. Add Preprocessing

Alternative 8: 2.9% accurate, 214.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 57.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 57.2

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 57.2

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

4. Applied rewrites 57.2%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 26.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 2.9%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))