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

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

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-122} \lor \neg \left(z \leq 2.15 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e-122) (not (<= z 2.15e-8)))
   (- (* x 1.0) (* z (sin y)))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-122) || !(z <= 2.15e-8)) {
		tmp = (x * 1.0) - (z * sin(y));
	} else {
		tmp = cos(y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.25d-122)) .or. (.not. (z <= 2.15d-8))) then
        tmp = (x * 1.0d0) - (z * sin(y))
    else
        tmp = cos(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-122) || !(z <= 2.15e-8)) {
		tmp = (x * 1.0) - (z * Math.sin(y));
	} else {
		tmp = Math.cos(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e-122) or not (z <= 2.15e-8):
		tmp = (x * 1.0) - (z * math.sin(y))
	else:
		tmp = math.cos(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e-122) || !(z <= 2.15e-8))
		tmp = Float64(Float64(x * 1.0) - Float64(z * sin(y)));
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.25e-122) || ~((z <= 2.15e-8)))
		tmp = (x * 1.0) - (z * sin(y));
	else
		tmp = cos(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e-122], N[Not[LessEqual[z, 2.15e-8]], $MachinePrecision]], N[(N[(x * 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e-122 or 2.1500000000000001e-8 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.7%

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

   if -1.25e-122 < z < 2.1500000000000001e-8

   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. 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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 - z \cdot \sin y}}{x}\right) \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 \cdot \frac{{z}^{2} \cdot {\sin y}^{2}}{x \cdot \cos y}} - z \cdot \sin y}{x}\right) \cdot x \]

      6. metadata-eval N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \cos y \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 88.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-122} \lor \neg \left(z \leq 2.15 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-122}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.25e-122)
   (- (* x 1.0) (* z (sin y)))
   (if (<= z 2.15e-8) (* (cos y) x) (fma (sin y) (- z) (* 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e-122) {
		tmp = (x * 1.0) - (z * sin(y));
	} else if (z <= 2.15e-8) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(sin(y), -z, (1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.25e-122)
		tmp = Float64(Float64(x * 1.0) - Float64(z * sin(y)));
	elseif (z <= 2.15e-8)
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(sin(y), Float64(-z), Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.25e-122], N[(N[(x * 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.15e-8], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e-122

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.0%

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

   if -1.25e-122 < z < 2.1500000000000001e-8

   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. 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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 - z \cdot \sin y}}{x}\right) \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 \cdot \frac{{z}^{2} \cdot {\sin y}^{2}}{x \cdot \cos y}} - z \cdot \sin y}{x}\right) \cdot x \]

      6. metadata-eval N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \cos y \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 88.3%

       \[\leadsto \cos y \cdot x \]

   if 2.1500000000000001e-8 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 88.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 88.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.1

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

   7. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-122}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+60} \lor \neg \left(z \leq 2.6 \cdot 10^{+130}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5e+60) (not (<= z 2.6e+130))) (* (- z) (sin y)) (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5e+60) || !(z <= 2.6e+130)) {
		tmp = -z * sin(y);
	} else {
		tmp = cos(y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-5d+60)) .or. (.not. (z <= 2.6d+130))) then
        tmp = -z * sin(y)
    else
        tmp = cos(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5e+60) || !(z <= 2.6e+130)) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = Math.cos(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5e+60) or not (z <= 2.6e+130):
		tmp = -z * math.sin(y)
	else:
		tmp = math.cos(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5e+60) || !(z <= 2.6e+130))
		tmp = Float64(Float64(-z) * sin(y));
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5e+60) || ~((z <= 2.6e+130)))
		tmp = -z * sin(y);
	else
		tmp = cos(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5e+60], N[Not[LessEqual[z, 2.6e+130]], $MachinePrecision]], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.99999999999999975e60 or 2.5999999999999998e130 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -4.99999999999999975e60 < z < 2.5999999999999998e130

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 - z \cdot \sin y}}{x}\right) \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 \cdot \frac{{z}^{2} \cdot {\sin y}^{2}}{x \cdot \cos y}} - z \cdot \sin y}{x}\right) \cdot x \]

      6. metadata-eval N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \cos y \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 80.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+60} \lor \neg \left(z \leq 2.6 \cdot 10^{+130}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.058 \lor \neg \left(y \leq 0.07\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.058) (not (<= y 0.07)))
   (* (cos y) x)
   (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.058) || !(y <= 0.07)) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.058) || !(y <= 0.07))
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.058], N[Not[LessEqual[y, 0.07]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0580000000000000029 or 0.070000000000000007 < 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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 - z \cdot \sin y}}{x}\right) \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 \cdot \frac{{z}^{2} \cdot {\sin y}^{2}}{x \cdot \cos y}} - z \cdot \sin y}{x}\right) \cdot x \]

      6. metadata-eval N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \cos y \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 50.5%

       \[\leadsto \cos y \cdot x \]

   if -0.0580000000000000029 < y < 0.070000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) \cdot y} + x \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.058 \lor \neg \left(y \leq 0.07\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-191} \lor \neg \left(x \leq 3.8 \cdot 10^{-155}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e-191) (not (<= x 3.8e-155))) (* 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-191) || !(x <= 3.8e-155)) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2d-191)) .or. (.not. (x <= 3.8d-155))) then
        tmp = 1.0d0 * x
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-191) || !(x <= 3.8e-155)) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e-191) or not (x <= 3.8e-155):
		tmp = 1.0 * x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e-191) || !(x <= 3.8e-155))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e-191) || ~((x <= 3.8e-155)))
		tmp = 1.0 * x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e-191], N[Not[LessEqual[x, 3.8e-155]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-191 or 3.7999999999999998e-155 < x

   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. 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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 - z \cdot \sin y}}{x}\right) \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \left(\cos y + \frac{\color{blue}{0 \cdot \frac{{z}^{2} \cdot {\sin y}^{2}}{x \cdot \cos y}} - z \cdot \sin y}{x}\right) \cdot x \]

      6. metadata-eval N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 44.8%

       \[\leadsto 1 \cdot x \]

   if -2e-191 < x < 3.7999999999999998e-155

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.7%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-191} \lor \neg \left(x \leq 3.8 \cdot 10^{-155}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.1

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

5. Applied rewrites 52.1%

   \[\leadsto \color{blue}{x - z \cdot y} \]
6. Add Preprocessing

Alternative 8: 38.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(1.0 * x), $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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.8

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. 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 x around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. neg-sub0 N/A

      \[\leadsto \left(\cos y + \frac{\color{blue}{0 - z \cdot \sin y}}{x}\right) \cdot x \]

   5. mul0-lft N/A

      \[\leadsto \left(\cos y + \frac{\color{blue}{0 \cdot \frac{{z}^{2} \cdot {\sin y}^{2}}{x \cdot \cos y}} - z \cdot \sin y}{x}\right) \cdot x \]

   6. metadata-eval N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 91.3%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 39.3%

    \[\leadsto 1 \cdot x \]

11. Final simplification 39.3%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))