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

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 74.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -8.4 \cdot 10^{+193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.065:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.032:\\ \;\;\;\;x + y \cdot \left(z + \left(z \cdot -2 + y \cdot \left(x \cdot -0.5 + \left(y \cdot z\right) \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* (sin y) (- z))))
   (if (<= y -8.4e+193)
     t_0
     (if (<= y -0.065)
       t_1
       (if (<= y 0.032)
         (+
          x
          (*
           y
           (+
            z
            (+
             (* z -2.0)
             (* y (+ (* x -0.5) (* (* y z) 0.16666666666666666)))))))
         (if (<= y 1.7e+250) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = sin(y) * -z;
	double tmp;
	if (y <= -8.4e+193) {
		tmp = t_0;
	} else if (y <= -0.065) {
		tmp = t_1;
	} else if (y <= 0.032) {
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))));
	} else if (y <= 1.7e+250) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * cos(y)
    t_1 = sin(y) * -z
    if (y <= (-8.4d+193)) then
        tmp = t_0
    else if (y <= (-0.065d0)) then
        tmp = t_1
    else if (y <= 0.032d0) then
        tmp = x + (y * (z + ((z * (-2.0d0)) + (y * ((x * (-0.5d0)) + ((y * z) * 0.16666666666666666d0))))))
    else if (y <= 1.7d+250) then
        tmp = t_1
    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 t_1 = Math.sin(y) * -z;
	double tmp;
	if (y <= -8.4e+193) {
		tmp = t_0;
	} else if (y <= -0.065) {
		tmp = t_1;
	} else if (y <= 0.032) {
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))));
	} else if (y <= 1.7e+250) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = math.sin(y) * -z
	tmp = 0
	if y <= -8.4e+193:
		tmp = t_0
	elif y <= -0.065:
		tmp = t_1
	elif y <= 0.032:
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))))
	elif y <= 1.7e+250:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (y <= -8.4e+193)
		tmp = t_0;
	elseif (y <= -0.065)
		tmp = t_1;
	elseif (y <= 0.032)
		tmp = Float64(x + Float64(y * Float64(z + Float64(Float64(z * -2.0) + Float64(y * Float64(Float64(x * -0.5) + Float64(Float64(y * z) * 0.16666666666666666)))))));
	elseif (y <= 1.7e+250)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	t_1 = sin(y) * -z;
	tmp = 0.0;
	if (y <= -8.4e+193)
		tmp = t_0;
	elseif (y <= -0.065)
		tmp = t_1;
	elseif (y <= 0.032)
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))));
	elseif (y <= 1.7e+250)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[y, -8.4e+193], t$95$0, If[LessEqual[y, -0.065], t$95$1, If[LessEqual[y, 0.032], N[(x + N[(y * N[(z + N[(N[(z * -2.0), $MachinePrecision] + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+250], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4e193 or 1.69999999999999987e250 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 68.1%

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

   if -8.4e193 < y < -0.065000000000000002 or 0.032000000000000001 < y < 1.69999999999999987e250

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 58.5%

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

      1. associate-*r* 58.5%

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

      2. neg-mul-1 58.5%

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

   5. Simplified 58.5%

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

   if -0.065000000000000002 < y < 0.032000000000000001

   1. Initial program 100.0%

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

      1. prod-diff 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. associate-+l+ 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-undefine 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 99.5%

      \[\leadsto \color{blue}{x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + y \cdot \left(-0.16666666666666666 \cdot z + 0.3333333333333333 \cdot z\right)\right)\right)\right)} \]

   6. Taylor expanded in z around 0 99.5%

      \[\leadsto x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + \color{blue}{0.16666666666666666 \cdot \left(y \cdot z\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + \color{blue}{\left(y \cdot z\right) \cdot 0.16666666666666666}\right)\right)\right) \]

      2. *-commutative 99.5%

         \[\leadsto x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + \color{blue}{\left(z \cdot y\right)} \cdot 0.16666666666666666\right)\right)\right) \]

   8. Simplified 99.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+193}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -0.065:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 0.032:\\ \;\;\;\;x + y \cdot \left(z + \left(z \cdot -2 + y \cdot \left(x \cdot -0.5 + \left(y \cdot z\right) \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+250}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0062 \lor \neg \left(y \leq 0.065\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + \left(z \cdot -2 + y \cdot \left(x \cdot -0.5 + \left(y \cdot z\right) \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0062) (not (<= y 0.065)))
   (* x (cos y))
   (+
    x
    (*
     y
     (+
      z
      (+ (* z -2.0) (* y (+ (* x -0.5) (* (* y z) 0.16666666666666666)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0062) || !(y <= 0.065)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))));
	}
	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 ((y <= (-0.0062d0)) .or. (.not. (y <= 0.065d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + ((z * (-2.0d0)) + (y * ((x * (-0.5d0)) + ((y * z) * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0062) || !(y <= 0.065)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0062) or not (y <= 0.065):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0062) || !(y <= 0.065))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(Float64(z * -2.0) + Float64(y * Float64(Float64(x * -0.5) + Float64(Float64(y * z) * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0062) || ~((y <= 0.065)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + ((z * -2.0) + (y * ((x * -0.5) + ((y * z) * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0062], N[Not[LessEqual[y, 0.065]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(N[(z * -2.0), $MachinePrecision] + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00619999999999999978 or 0.065000000000000002 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 49.7%

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

   if -0.00619999999999999978 < y < 0.065000000000000002

   1. Initial program 100.0%

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

      1. prod-diff 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. associate-+l+ 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-undefine 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + y \cdot \left(-0.16666666666666666 \cdot z + 0.3333333333333333 \cdot z\right)\right)\right)\right)} \]

   6. Taylor expanded in z around 0 100.0%

      \[\leadsto x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + \color{blue}{0.16666666666666666 \cdot \left(y \cdot z\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + \color{blue}{\left(y \cdot z\right) \cdot 0.16666666666666666}\right)\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto x + y \cdot \left(z + \left(-2 \cdot z + y \cdot \left(-0.5 \cdot x + \color{blue}{\left(z \cdot y\right)} \cdot 0.16666666666666666\right)\right)\right) \]

   8. Simplified 100.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0062 \lor \neg \left(y \leq 0.065\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + \left(z \cdot -2 + y \cdot \left(x \cdot -0.5 + \left(y \cdot z\right) \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.8% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.95e-25) x (if (<= x 3.15e-99) (* y (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e-25) {
		tmp = x;
	} else if (x <= 3.15e-99) {
		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 (x <= (-1.95d-25)) then
        tmp = x
    else if (x <= 3.15d-99) 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 (x <= -1.95e-25) {
		tmp = x;
	} else if (x <= 3.15e-99) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.95e-25:
		tmp = x
	elif x <= 3.15e-99:
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.95e-25)
		tmp = x;
	elseif (x <= 3.15e-99)
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.95e-25)
		tmp = x;
	elseif (x <= 3.15e-99)
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.95e-25], x, If[LessEqual[x, 3.15e-99], N[(y * (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e-25 or 3.14999999999999996e-99 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 46.6%

      \[\leadsto \color{blue}{x} \]

   if -1.95e-25 < x < 3.14999999999999996e-99

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 75.6%

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

      1. associate-*r* 75.6%

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

      2. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in y around 0 34.7%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - N[(y * z), $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 51.1%

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

   1. mul-1-neg 51.1%

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

   2. unsub-neg 51.1%

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

   3. *-commutative 51.1%

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

5. Simplified 51.1%

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

6. Final simplification 51.1%

   \[\leadsto x - y \cdot z \]
7. Add Preprocessing

Alternative 6: 38.0% accurate, 207.0× 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 36.7%

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

Reproduce

herbie shell --seed 2024137 
(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))))