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

Percentage Accurate: 99.8% → 99.8%

Time: 6.9s

Alternatives: 7

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: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-46} \lor \neg \left(z \leq 2.25 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6e-46) (not (<= z 2.25e-68)))
   (- (* x 1.0) (* z (sin y)))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e-46) || !(z <= 2.25e-68)) {
		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 <= (-6d-46)) .or. (.not. (z <= 2.25d-68))) 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 <= -6e-46) || !(z <= 2.25e-68)) {
		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 <= -6e-46) or not (z <= 2.25e-68):
		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 <= -6e-46) || !(z <= 2.25e-68))
		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 <= -6e-46) || ~((z <= 2.25e-68)))
		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, -6e-46], N[Not[LessEqual[z, 2.25e-68]], $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 < -5.99999999999999975e-46 or 2.25e-68 < 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 90.0%

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

   if -5.99999999999999975e-46 < z < 2.25e-68

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. remove-double-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-in N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-46} \lor \neg \left(z \leq 2.25 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+160} \lor \neg \left(z \leq 7.2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.7e+160) (not (<= z 7.2e+87)))
   (* (- z) (sin y))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.7e+160) || !(z <= 7.2e+87)) {
		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 <= (-4.7d+160)) .or. (.not. (z <= 7.2d+87))) 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 <= -4.7e+160) || !(z <= 7.2e+87)) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = Math.cos(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.7e+160) or not (z <= 7.2e+87):
		tmp = -z * math.sin(y)
	else:
		tmp = math.cos(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.7e+160) || !(z <= 7.2e+87))
		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 <= -4.7e+160) || ~((z <= 7.2e+87)))
		tmp = -z * sin(y);
	else
		tmp = cos(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.7e+160], N[Not[LessEqual[z, 7.2e+87]], $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 < -4.6999999999999997e160 or 7.19999999999999988e87 < 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 76.8

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

   5. Applied rewrites 76.8%

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

   if -4.6999999999999997e160 < z < 7.19999999999999988e87

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. remove-double-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-in N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+160} \lor \neg \left(z \leq 7.2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.042 \lor \neg \left(y \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.042) (not (<= y 1.3e-7)))
   (* (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 <= -0.042) || !(y <= 1.3e-7)) {
		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 <= -0.042) || !(y <= 1.3e-7))
		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, -0.042], N[Not[LessEqual[y, 1.3e-7]], $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 < -0.0420000000000000026 or 1.29999999999999999e-7 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. remove-double-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-in N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.8%

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

   if -0.0420000000000000026 < y < 1.29999999999999999e-7

   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.042 \lor \neg \left(y \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -5e+164) (* (- y) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+164) {
		tmp = -y * z;
	} else {
		tmp = 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+164)) then
        tmp = -y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+164) {
		tmp = -y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e+164:
		tmp = -y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+164)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e+164)
		tmp = -y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e+164], N[((-y) * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999995e164

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 49.7

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

   5. Applied rewrites 49.7%

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

   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 44.8%

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

   if -4.9999999999999995e164 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 5.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 4.8%

       \[\leadsto -x \]
   11. Step-by-step derivation

   12. Applied rewrites 43.5%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.6% accurate, 23.8× 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(Float64(-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

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 51.6

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

5. Applied rewrites 51.6%

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

Alternative 7: 38.6% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

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. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 51.6

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

5. Applied rewrites 51.6%

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 4.5%

    \[\leadsto -x \]
11. Step-by-step derivation

12. Applied rewrites 39.8%

    \[\leadsto \color{blue}{x} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))))