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

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. 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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

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 3: 85.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-38}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.7e+67) t_0 (if (<= x 1.1e-38) (- x (* z (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.7e+67) {
		tmp = t_0;
	} else if (x <= 1.1e-38) {
		tmp = x - (z * sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (x <= (-1.7d+67)) then
        tmp = t_0
    else if (x <= 1.1d-38) then
        tmp = x - (z * sin(y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (x <= -1.7e+67) {
		tmp = t_0;
	} else if (x <= 1.1e-38) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.7e+67:
		tmp = t_0
	elif x <= 1.1e-38:
		tmp = x - (z * math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.7e+67)
		tmp = t_0;
	elseif (x <= 1.1e-38)
		tmp = Float64(x - Float64(z * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -1.7e+67)
		tmp = t_0;
	elseif (x <= 1.1e-38)
		tmp = x - (z * sin(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+67], t$95$0, If[LessEqual[x, 1.1e-38], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7000000000000001e67 or 1.10000000000000004e-38 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.2%

      \[\leadsto \mathsf{fma}\left(-0.5, y \cdot y, 1\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 87.9

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

   11. Applied rewrites 87.9%

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

   if -1.7000000000000001e67 < x < 1.10000000000000004e-38

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. div-inv N/A

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

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

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. sin-neg-rev N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-rgt-identity-rev N/A

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

      10. sin-PI/2 N/A

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

      11. *-rgt-identity 86.4

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

   7. Applied rewrites 86.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-38}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.07:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.07)
     t_0
     (if (<= y 0.135)
       (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.07) {
		tmp = t_0;
	} else if (y <= 0.135) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.07)
		tmp = t_0;
	elseif (y <= 0.135)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.07], t$95$0, If[LessEqual[y, 0.135], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.070000000000000007 or 0.13500000000000001 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left(-0.5, y \cdot y, 1\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 55.1

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

   11. Applied rewrites 55.1%

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

   if -0.070000000000000007 < y < 0.13500000000000001

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.07:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% 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 54.1

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

5. Applied rewrites 54.1%

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

Alternative 6: 17.1% accurate, 26.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
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 54.1

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

5. Applied rewrites 54.1%

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

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 16.9%

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

Reproduce

herbie shell --seed 2024298 
(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))))