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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

Alternatives: 6

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: 86.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-143} \lor \neg \left(z \leq 1.52 \cdot 10^{-99}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.8e-143) (not (<= z 1.52e-99)))
   (- (* x 1.0) (* z (sin y)))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.8e-143) || !(z <= 1.52e-99)) {
		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 <= (-7.8d-143)) .or. (.not. (z <= 1.52d-99))) 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 <= -7.8e-143) || !(z <= 1.52e-99)) {
		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 <= -7.8e-143) or not (z <= 1.52e-99):
		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 <= -7.8e-143) || !(z <= 1.52e-99))
		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 <= -7.8e-143) || ~((z <= 1.52e-99)))
		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, -7.8e-143], N[Not[LessEqual[z, 1.52e-99]], $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 < -7.80000000000000007e-143 or 1.51999999999999999e-99 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 85.6%

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

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

   1. Initial program 99.8%

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

   3. 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 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 94.8%

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

4. Final simplification 88.5%

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

Alternative 3: 73.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

   3. 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 92.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.5%

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

   if -1.64999999999999996e171 < y < -0.839999999999999969

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 60.6

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

   5. Applied rewrites 60.6%

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

   if -0.839999999999999969 < y < 7.99999999999999956e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 4: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{-5} \lor \neg \left(y \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.75e-5) (not (<= y 8e-20))) (* (cos y) x) (fma (- z) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.75e-5) || !(y <= 8e-20)) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(-z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.75e-5) || !(y <= 8e-20))
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(Float64(-z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.75e-5], N[Not[LessEqual[y, 8e-20]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[((-z) * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7500000000000001e-5 or 7.99999999999999956e-20 < y

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 89.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.3%

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

   if -2.7500000000000001e-5 < y < 7.99999999999999956e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{-5} \lor \neg \left(y \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% 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 54.0

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

5. Applied rewrites 54.0%

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

Alternative 6: 16.8% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \left(-y\right) \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[((-y) * z), $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 54.0

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

5. Applied rewrites 54.0%

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

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 17.4%

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

Reproduce

herbie shell --seed 2024332 
(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))))