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

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 74.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ t_1 := \cos y \cdot x\\ \mathbf{if}\;y \leq -4.05 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.034:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))) (t_1 (* (cos y) x)))
   (if (<= y -4.05e+96)
     t_0
     (if (<= y -0.034)
       t_1
       (if (<= y 2.5e-12)
         (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
         (if (<= y 6e+120) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double t_1 = cos(y) * x;
	double tmp;
	if (y <= -4.05e+96) {
		tmp = t_0;
	} else if (y <= -0.034) {
		tmp = t_1;
	} else if (y <= 2.5e-12) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else if (y <= 6e+120) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	t_1 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -4.05e+96)
		tmp = t_0;
	elseif (y <= -0.034)
		tmp = t_1;
	elseif (y <= 2.5e-12)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	elseif (y <= 6e+120)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -4.05e+96], t$95$0, If[LessEqual[y, -0.034], t$95$1, If[LessEqual[y, 2.5e-12], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 6e+120], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0500000000000001e96 or 6e120 < y

   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 62.0

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

   5. Applied rewrites 62.0%

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

   if -4.0500000000000001e96 < y < -0.034000000000000002 or 2.49999999999999985e-12 < y < 6e120

   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 55.2

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

   4. Applied rewrites 55.2%

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

   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. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 67.3

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

   7. Applied rewrites 67.3%

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

   if -0.034000000000000002 < y < 2.49999999999999985e-12

   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 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.05 \cdot 10^{+96}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq -0.034:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+120}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+56} \lor \neg \left(x \leq 2.2 \cdot 10^{+165}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.9e+56) (not (<= x 2.2e+165)))
   (* (cos y) x)
   (- (* x 1.0) (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e+56) || !(x <= 2.2e+165)) {
		tmp = cos(y) * x;
	} else {
		tmp = (x * 1.0) - (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 <= (-3.9d+56)) .or. (.not. (x <= 2.2d+165))) then
        tmp = cos(y) * x
    else
        tmp = (x * 1.0d0) - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e+56) || !(x <= 2.2e+165)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = (x * 1.0) - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.9e+56) or not (x <= 2.2e+165):
		tmp = math.cos(y) * x
	else:
		tmp = (x * 1.0) - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.9e+56) || !(x <= 2.2e+165))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(Float64(x * 1.0) - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.9e+56) || ~((x <= 2.2e+165)))
		tmp = cos(y) * x;
	else
		tmp = (x * 1.0) - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.9e+56], N[Not[LessEqual[x, 2.2e+165]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.89999999999999994e56 or 2.1999999999999999e165 < 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 49.1

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

   4. Applied rewrites 49.1%

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

   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. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 91.6

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

   7. Applied rewrites 91.6%

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

   if -3.89999999999999994e56 < x < 2.1999999999999999e165

   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 86.0%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+56} \lor \neg \left(x \leq 2.2 \cdot 10^{+165}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+56}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+146}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e+56)
   (* (cos y) x)
   (if (<= x 1.55e+146)
     (- (* x 1.0) (* z (sin y)))
     (- (* x (cos y)) (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e+56) {
		tmp = cos(y) * x;
	} else if (x <= 1.55e+146) {
		tmp = (x * 1.0) - (z * sin(y));
	} else {
		tmp = (x * cos(y)) - (z * 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 <= (-3.9d+56)) then
        tmp = cos(y) * x
    else if (x <= 1.55d+146) then
        tmp = (x * 1.0d0) - (z * sin(y))
    else
        tmp = (x * cos(y)) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e+56) {
		tmp = Math.cos(y) * x;
	} else if (x <= 1.55e+146) {
		tmp = (x * 1.0) - (z * Math.sin(y));
	} else {
		tmp = (x * Math.cos(y)) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.9e+56:
		tmp = math.cos(y) * x
	elif x <= 1.55e+146:
		tmp = (x * 1.0) - (z * math.sin(y))
	else:
		tmp = (x * math.cos(y)) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e+56)
		tmp = Float64(cos(y) * x);
	elseif (x <= 1.55e+146)
		tmp = Float64(Float64(x * 1.0) - Float64(z * sin(y)));
	else
		tmp = Float64(Float64(x * cos(y)) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.9e+56)
		tmp = cos(y) * x;
	elseif (x <= 1.55e+146)
		tmp = (x * 1.0) - (z * sin(y));
	else
		tmp = (x * cos(y)) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e+56], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.55e+146], N[(N[(x * 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999994e56

   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 32.5

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

   4. Applied rewrites 32.5%

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

   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. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 89.5

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

   7. Applied rewrites 89.5%

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

   if -3.89999999999999994e56 < x < 1.5500000000000001e146

   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 86.4%

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

   if 1.5500000000000001e146 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+56}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+146}:\\ \;\;\;\;x \cdot 1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.034 \lor \neg \left(y \leq 2.5 \cdot 10^{-12}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.034) (not (<= y 2.5e-12)))
   (* (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 <= -0.034) || !(y <= 2.5e-12)) {
		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 <= -0.034) || !(y <= 2.5e-12))
		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, -0.034], N[Not[LessEqual[y, 2.5e-12]], $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 < -0.034000000000000002 or 2.49999999999999985e-12 < 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 54.0

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

   4. Applied rewrites 54.0%

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

   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. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 50.2

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

   7. Applied rewrites 50.2%

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

   if -0.034000000000000002 < y < 2.49999999999999985e-12

   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 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.034 \lor \neg \left(y \leq 2.5 \cdot 10^{-12}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+118} \lor \neg \left(z \leq 3.7 \cdot 10^{+111}\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e+118) (not (<= z 3.7e+111))) (* (- y) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+118) || !(z <= 3.7e+111)) {
		tmp = -y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2d+118)) .or. (.not. (z <= 3.7d+111))) then
        tmp = -y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+118) || !(z <= 3.7e+111)) {
		tmp = -y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2e+118) or not (z <= 3.7e+111):
		tmp = -y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e+118) || !(z <= 3.7e+111))
		tmp = Float64(Float64(-y) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2e+118) || ~((z <= 3.7e+111)))
		tmp = -y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e+118], N[Not[LessEqual[z, 3.7e+111]], $MachinePrecision]], N[((-y) * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999993e118 or 3.7000000000000003e111 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 50.8

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

   5. Applied rewrites 50.8%

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

   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 35.8%

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

   if -1.99999999999999993e118 < z < 3.7000000000000003e111

   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 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 31.6%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 51.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+118} \lor \neg \left(z \leq 3.7 \cdot 10^{+111}\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 53.9

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

5. Applied rewrites 53.9%

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

Alternative 8: 39.0% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 29.1%

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

8. Taylor expanded in x around -inf

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

10. Applied rewrites 41.0%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))