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

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

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: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-176} \lor \neg \left(z \leq 1.35 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.45e-176) (not (<= z 1.35e-35)))
   (- (* x 1.0) (* z (sin y)))
   (* (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.45e-176) || !(z <= 1.35e-35)) {
		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 <= (-1.45d-176)) .or. (.not. (z <= 1.35d-35))) 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 <= -1.45e-176) || !(z <= 1.35e-35)) {
		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 <= -1.45e-176) or not (z <= 1.35e-35):
		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 <= -1.45e-176) || !(z <= 1.35e-35))
		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 <= -1.45e-176) || ~((z <= 1.35e-35)))
		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, -1.45e-176], N[Not[LessEqual[z, 1.35e-35]], $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 < -1.45000000000000003e-176 or 1.3499999999999999e-35 < 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 87.4%

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

   if -1.45000000000000003e-176 < z < 1.3499999999999999e-35

   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. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \cos y}\right)\right) \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-176} \lor \neg \left(z \leq 1.35 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.43 \lor \neg \left(x \leq 1.6 \cdot 10^{-110}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.43) (not (<= x 1.6e-110))) (* (cos y) x) (* (- z) (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.43) || !(x <= 1.6e-110)) {
		tmp = cos(y) * x;
	} else {
		tmp = -z * sin(y);
	}
	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 <= (-0.43d0)) .or. (.not. (x <= 1.6d-110))) then
        tmp = cos(y) * x
    else
        tmp = -z * sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.43) || !(x <= 1.6e-110)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = -z * Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.43) or not (x <= 1.6e-110):
		tmp = math.cos(y) * x
	else:
		tmp = -z * math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.43) || !(x <= 1.6e-110))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(Float64(-z) * sin(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.43) || ~((x <= 1.6e-110)))
		tmp = cos(y) * x;
	else
		tmp = -z * sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.43], N[Not[LessEqual[x, 1.6e-110]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.429999999999999993 or 1.60000000000000014e-110 < x

   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. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \cos y}\right)\right) \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if -0.429999999999999993 < x < 1.60000000000000014e-110

   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 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.43 \lor \neg \left(x \leq 1.6 \cdot 10^{-110}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.45 \lor \neg \left(y \leq 920000\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.45) (not (<= y 920000.0)))
   (* (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 <= -0.45) || !(y <= 920000.0)) {
		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 <= -0.45) || !(y <= 920000.0))
		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, -0.45], N[Not[LessEqual[y, 920000.0]], $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 < -0.450000000000000011 or 9.2e5 < 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. lift-*.f64 N/A

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

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

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(x \cdot \cos y\right)}^{2}}} + \left(\mathsf{neg}\left(z\right)\right) \cdot \sin y \]

      7. pow2 N/A

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

      8. sqrt-prod N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\cos y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \cos y}\right)\right) \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 50.0

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

   7. Applied rewrites 50.0%

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

   if -0.450000000000000011 < y < 9.2e5

   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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.45 \lor \neg \left(y \leq 920000\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-37} \lor \neg \left(x \leq 2.5 \cdot 10^{-171}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55e-37) (not (<= x 2.5e-171))) (* 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55e-37) || !(x <= 2.5e-171)) {
		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 <= (-1.55d-37)) .or. (.not. (x <= 2.5d-171))) 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 <= -1.55e-37) || !(x <= 2.5e-171)) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.55e-37) or not (x <= 2.5e-171):
		tmp = 1.0 * x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.55e-37) || !(x <= 2.5e-171))
		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 <= -1.55e-37) || ~((x <= 2.5e-171)))
		tmp = 1.0 * x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.55e-37], N[Not[LessEqual[x, 2.5e-171]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999997e-37 or 2.49999999999999996e-171 < x

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 98.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 48.8%

      \[\leadsto 1 \cdot x \]

   if -1.54999999999999997e-37 < x < 2.49999999999999996e-171

   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 54.4

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

   5. Applied rewrites 54.4%

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

   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 40.0%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-37} \lor \neg \left(x \leq 2.5 \cdot 10^{-171}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% 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 52.9

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

5. Applied rewrites 52.9%

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

6. Final simplification 52.9%

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

Alternative 7: 38.4% 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. 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 91.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 39.2%

   \[\leadsto 1 \cdot x \]

9. Final simplification 39.2%

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

Reproduce

herbie shell --seed 2024337 
(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))))