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

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] * x + N[(z * N[Cos[y], $MachinePrecision]), $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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 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-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.27:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), x, \mathsf{fma}\left(z, \left(y \cdot y\right) \cdot -0.5, z\right)\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -0.32)
     t_0
     (if (<= y 0.27)
       (fma
        (fma
         (fma
          (* y y)
          (fma (* y y) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666)
         (* y (* y y))
         y)
        x
        (fma z (* (* y y) -0.5) z))
       (if (<= y 3.7e+106) t_0 (* (sin y) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -0.32) {
		tmp = t_0;
	} else if (y <= 0.27) {
		tmp = fma(fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y), x, fma(z, ((y * y) * -0.5), z));
	} else if (y <= 3.7e+106) {
		tmp = t_0;
	} else {
		tmp = sin(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -0.32)
		tmp = t_0;
	elseif (y <= 0.27)
		tmp = fma(fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y), x, fma(z, Float64(Float64(y * y) * -0.5), z));
	elseif (y <= 3.7e+106)
		tmp = t_0;
	else
		tmp = Float64(sin(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.32], t$95$0, If[LessEqual[y, 0.27], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * x + N[(z * N[(N[(y * y), $MachinePrecision] * -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+106], t$95$0, N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.320000000000000007 or 0.27000000000000002 < y < 3.69999999999999995e106

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 60.5

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

   5. Simplified 60.5%

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

   if -0.320000000000000007 < y < 0.27000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y + 1 \cdot y\right)} + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y\right)} + 1 \cdot y\right) + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      4. *-lft-identity N/A

         \[\leadsto x \cdot \left({y}^{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y\right) + \color{blue}{y}\right) + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y, y\right)} + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

   8. Simplified 99.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right) \cdot y, y\right)} + \mathsf{fma}\left(z, -0.5 \cdot \left(y \cdot y\right), z\right) \]

   10. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), x, \mathsf{fma}\left(z, \left(y \cdot y\right) \cdot -0.5, z\right)\right)} \]

   if 3.69999999999999995e106 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 59.3

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

   5. Simplified 59.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.27:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), x, \mathsf{fma}\left(z, \left(y \cdot y\right) \cdot -0.5, z\right)\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+106}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-64}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -5.2e+27) t_0 (if (<= z 1.15e-64) (+ z (* (sin y) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -5.2e+27) {
		tmp = t_0;
	} else if (z <= 1.15e-64) {
		tmp = z + (sin(y) * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * cos(y)
    if (z <= (-5.2d+27)) then
        tmp = t_0
    else if (z <= 1.15d-64) then
        tmp = z + (sin(y) * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -5.2e+27) {
		tmp = t_0;
	} else if (z <= 1.15e-64) {
		tmp = z + (Math.sin(y) * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -5.2e+27:
		tmp = t_0
	elif z <= 1.15e-64:
		tmp = z + (math.sin(y) * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -5.2e+27)
		tmp = t_0;
	elseif (z <= 1.15e-64)
		tmp = Float64(z + Float64(sin(y) * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -5.2e+27)
		tmp = t_0;
	elseif (z <= 1.15e-64)
		tmp = z + (sin(y) * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+27], t$95$0, If[LessEqual[z, 1.15e-64], N[(z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018e27 or 1.1500000000000001e-64 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.2

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

   5. Simplified 82.2%

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

   if -5.20000000000000018e27 < z < 1.1500000000000001e-64

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.0%

      \[\leadsto x \cdot \sin y + z \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+27}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-64}:\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -5.2e+27) t_0 (if (<= z 1.15e-64) (fma (sin y) x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -5.2e+27) {
		tmp = t_0;
	} else if (z <= 1.15e-64) {
		tmp = fma(sin(y), x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -5.2e+27)
		tmp = t_0;
	elseif (z <= 1.15e-64)
		tmp = fma(sin(y), x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+27], t$95$0, If[LessEqual[z, 1.15e-64], N[(N[Sin[y], $MachinePrecision] * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018e27 or 1.1500000000000001e-64 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.2

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

   5. Simplified 82.2%

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

   if -5.20000000000000018e27 < z < 1.1500000000000001e-64

   1. Initial program 99.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      11. lower-/.f64 99.6

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \sin y + z \cdot \cos y}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y + x \cdot \sin y}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y} + x \cdot \sin y}} \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 91.7%

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

      1. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{z \cdot 1 + x \cdot \color{blue}{\sin y}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{z \cdot 1 + \color{blue}{x \cdot \sin y}}} \]

      3. lift-fma.f64 N/A

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

      4. remove-double-div 92.0

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

      5. lift-fma.f64 N/A

         \[\leadsto \color{blue}{z \cdot 1 + x \cdot \sin y} \]

      6. *-rgt-identity N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 92.0

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

   9. Applied egg-rr 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, x, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), x, \mathsf{fma}\left(z, \left(y \cdot y\right) \cdot -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -0.39)
     t_0
     (if (<= y 4.5e+16)
       (fma
        (fma
         (fma
          (* y y)
          (fma (* y y) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666)
         (* y (* y y))
         y)
        x
        (fma z (* (* y y) -0.5) z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -0.39) {
		tmp = t_0;
	} else if (y <= 4.5e+16) {
		tmp = fma(fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y), x, fma(z, ((y * y) * -0.5), z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -0.39)
		tmp = t_0;
	elseif (y <= 4.5e+16)
		tmp = fma(fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y), x, fma(z, Float64(Float64(y * y) * -0.5), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.39], t$95$0, If[LessEqual[y, 4.5e+16], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * x + N[(z * N[(N[(y * y), $MachinePrecision] * -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.39000000000000001 or 4.5e16 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.1

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

   5. Simplified 47.1%

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

   if -0.39000000000000001 < y < 4.5e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y + 1 \cdot y\right)} + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y\right)} + 1 \cdot y\right) + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      4. *-lft-identity N/A

         \[\leadsto x \cdot \left({y}^{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y\right) + \color{blue}{y}\right) + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y, y\right)} + \mathsf{fma}\left(z, \frac{-1}{2} \cdot \left(y \cdot y\right), z\right) \]

   8. Simplified 98.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right) \cdot y, y\right)} + \mathsf{fma}\left(z, -0.5 \cdot \left(y \cdot y\right), z\right) \]

   10. Applied egg-rr 98.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), x, \mathsf{fma}\left(z, \left(y \cdot y\right) \cdot -0.5, z\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), x, \mathsf{fma}\left(z, \left(y \cdot y\right) \cdot -0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+139}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -6e+139) (* y x) z))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e+139) {
		tmp = y * x;
	} else {
		tmp = 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 <= (-6d+139)) then
        tmp = y * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e+139) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6e+139:
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6e+139)
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6e+139)
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6e+139], N[(y * x), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -5.9999999999999999e139

   1. Initial program 99.6%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 42.9

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

   8. Simplified 42.9%

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

   if -5.9999999999999999e139 < x

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      11. lower-/.f64 99.5

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \sin y + z \cdot \cos y}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y + x \cdot \sin y}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y} + x \cdot \sin y}} \]

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 45.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z}}} \]

   7. Simplified 45.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z}}} \]
   8. Step-by-step derivation

      1. remove-double-div 45.3

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

   9. Applied egg-rr 45.3%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+139}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.3% 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 55.6

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

5. Simplified 55.6%

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

Alternative 9: 39.0% accurate, 214.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. flip-+ N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip-+ N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

   11. lower-/.f64 99.5

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \sin y + z \cdot \cos y}}} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y + x \cdot \sin y}}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y} + x \cdot \sin y}} \]

4. Applied egg-rr 99.5%

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

5. Taylor expanded in y around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{z}}} \]
6. Step-by-step derivation

   1. lower-/.f64 39.9

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z}}} \]

7. Simplified 39.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{z}}} \]
8. Step-by-step derivation

   1. remove-double-div 40.0

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

9. Applied egg-rr 40.0%

   \[\leadsto \color{blue}{z} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))))