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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 10

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. cancel-sign-sub-inv 99.7%

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

   2. +-commutative 99.7%

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

   3. distribute-lft-neg-out 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

   5. sin-neg 99.7%

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

   6. fma-define 99.7%

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

   7. sin-neg 99.7%

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

3. Simplified 99.7%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

Alternative 3: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+183} \lor \neg \left(x \leq -2.65 \cdot 10^{+129}\right) \land \left(x \leq -2.6 \cdot 10^{+22} \lor \neg \left(x \leq 7.4 \cdot 10^{-13}\right)\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.5e+183)
         (and (not (<= x -2.65e+129))
              (or (<= x -2.6e+22) (not (<= x 7.4e-13)))))
   (* x (cos y))
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e+183) || (!(x <= -2.65e+129) && ((x <= -2.6e+22) || !(x <= 7.4e-13)))) {
		tmp = x * cos(y);
	} else {
		tmp = x - (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 <= (-9.5d+183)) .or. (.not. (x <= (-2.65d+129))) .and. (x <= (-2.6d+22)) .or. (.not. (x <= 7.4d-13))) then
        tmp = x * cos(y)
    else
        tmp = x - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e+183) || (!(x <= -2.65e+129) && ((x <= -2.6e+22) || !(x <= 7.4e-13)))) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.5e+183) or (not (x <= -2.65e+129) and ((x <= -2.6e+22) or not (x <= 7.4e-13))):
		tmp = x * math.cos(y)
	else:
		tmp = x - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.5e+183) || (!(x <= -2.65e+129) && ((x <= -2.6e+22) || !(x <= 7.4e-13))))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.5e+183) || (~((x <= -2.65e+129)) && ((x <= -2.6e+22) || ~((x <= 7.4e-13)))))
		tmp = x * cos(y);
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.5e+183], And[N[Not[LessEqual[x, -2.65e+129]], $MachinePrecision], Or[LessEqual[x, -2.6e+22], N[Not[LessEqual[x, 7.4e-13]], $MachinePrecision]]]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000003e183 or -2.6499999999999999e129 < x < -2.6e22 or 7.39999999999999977e-13 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 91.7%

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

   if -9.5000000000000003e183 < x < -2.6499999999999999e129 or -2.6e22 < x < 7.39999999999999977e-13

   1. Initial program 99.7%

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

      1. add-cbrt-cube 64.7%

         \[\leadsto x \cdot \cos y - \color{blue}{\sqrt[3]{\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right) \cdot \left(z \cdot \sin y\right)}} \]

      2. pow3 64.7%

         \[\leadsto x \cdot \cos y - \sqrt[3]{\color{blue}{{\left(z \cdot \sin y\right)}^{3}}} \]

   4. Applied egg-rr 64.7%

      \[\leadsto x \cdot \cos y - \color{blue}{\sqrt[3]{{\left(z \cdot \sin y\right)}^{3}}} \]

   5. Taylor expanded in y around 0 59.7%

      \[\leadsto \color{blue}{x} - \sqrt[3]{{\left(z \cdot \sin y\right)}^{3}} \]
   6. Step-by-step derivation

      1. rem-cbrt-cube 93.6%

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

      2. *-commutative 93.6%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+183} \lor \neg \left(x \leq -2.65 \cdot 10^{+129}\right) \land \left(x \leq -2.6 \cdot 10^{+22} \lor \neg \left(x \leq 7.4 \cdot 10^{-13}\right)\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+183} \lor \neg \left(x \leq -5 \cdot 10^{+135} \lor \neg \left(x \leq -1.75 \cdot 10^{+21}\right) \land x \leq 10^{-11}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+183)
         (not (or (<= x -5e+135) (and (not (<= x -1.75e+21)) (<= x 1e-11)))))
   (* x (cos y))
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+183) || !((x <= -5e+135) || (!(x <= -1.75e+21) && (x <= 1e-11)))) {
		tmp = x * cos(y);
	} else {
		tmp = x - (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 <= (-6.8d+183)) .or. (.not. (x <= (-5d+135)) .or. (.not. (x <= (-1.75d+21))) .and. (x <= 1d-11))) then
        tmp = x * cos(y)
    else
        tmp = x - (z * sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+183) || !((x <= -5e+135) || (!(x <= -1.75e+21) && (x <= 1e-11)))) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e+183) or not ((x <= -5e+135) or (not (x <= -1.75e+21) and (x <= 1e-11))):
		tmp = x * math.cos(y)
	else:
		tmp = x - (z * math.sin(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e+183) || !((x <= -5e+135) || (!(x <= -1.75e+21) && (x <= 1e-11))))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(z * sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.8e+183) || ~(((x <= -5e+135) || (~((x <= -1.75e+21)) && (x <= 1e-11)))))
		tmp = x * cos(y);
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e+183], N[Not[Or[LessEqual[x, -5e+135], And[N[Not[LessEqual[x, -1.75e+21]], $MachinePrecision], LessEqual[x, 1e-11]]]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e183 or -5.00000000000000029e135 < x < -1.75e21 or 9.99999999999999939e-12 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 91.7%

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

   if -6.8e183 < x < -5.00000000000000029e135 or -1.75e21 < x < 9.99999999999999939e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 93.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+183} \lor \neg \left(x \leq -5 \cdot 10^{+135} \lor \neg \left(x \leq -1.75 \cdot 10^{+21}\right) \land x \leq 10^{-11}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* z (- (sin y)))))
   (if (<= x -4.5e-29)
     t_0
     (if (<= x -4.4e-92)
       t_1
       (if (<= x -2e-141) (fma (- y) z x) (if (<= x 2.1e-108) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = z * -sin(y);
	double tmp;
	if (x <= -4.5e-29) {
		tmp = t_0;
	} else if (x <= -4.4e-92) {
		tmp = t_1;
	} else if (x <= -2e-141) {
		tmp = fma(-y, z, x);
	} else if (x <= 2.1e-108) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(z * Float64(-sin(y)))
	tmp = 0.0
	if (x <= -4.5e-29)
		tmp = t_0;
	elseif (x <= -4.4e-92)
		tmp = t_1;
	elseif (x <= -2e-141)
		tmp = fma(Float64(-y), z, x);
	elseif (x <= 2.1e-108)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[x, -4.5e-29], t$95$0, If[LessEqual[x, -4.4e-92], t$95$1, If[LessEqual[x, -2e-141], N[((-y) * z + x), $MachinePrecision], If[LessEqual[x, 2.1e-108], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4999999999999998e-29 or 2.0999999999999999e-108 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 82.9%

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

   if -4.4999999999999998e-29 < x < -4.39999999999999974e-92 or -2.0000000000000001e-141 < x < 2.0999999999999999e-108

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 76.5%

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

      1. neg-mul-1 76.5%

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

      2. distribute-rgt-neg-in 76.5%

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

   5. Simplified 76.5%

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

   if -4.39999999999999974e-92 < x < -2.0000000000000001e-141

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. associate-*r* 84.5%

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

      3. fma-define 84.5%

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

      4. mul-1-neg 84.5%

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

   5. Simplified 84.5%

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

Alternative 6: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-154}:\\ \;\;\;\;x - z \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* z (- (sin y)))))
   (if (<= x -1.45e-28)
     t_0
     (if (<= x -4.8e-92)
       t_1
       (if (<= x -6.2e-154) (- x (* z y)) (if (<= x 3.3e-109) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double t_1 = z * -sin(y);
	double tmp;
	if (x <= -1.45e-28) {
		tmp = t_0;
	} else if (x <= -4.8e-92) {
		tmp = t_1;
	} else if (x <= -6.2e-154) {
		tmp = x - (z * y);
	} else if (x <= 3.3e-109) {
		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 = z * -sin(y)
    if (x <= (-1.45d-28)) then
        tmp = t_0
    else if (x <= (-4.8d-92)) then
        tmp = t_1
    else if (x <= (-6.2d-154)) then
        tmp = x - (z * y)
    else if (x <= 3.3d-109) 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 = z * -Math.sin(y);
	double tmp;
	if (x <= -1.45e-28) {
		tmp = t_0;
	} else if (x <= -4.8e-92) {
		tmp = t_1;
	} else if (x <= -6.2e-154) {
		tmp = x - (z * y);
	} else if (x <= 3.3e-109) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = z * -math.sin(y)
	tmp = 0
	if x <= -1.45e-28:
		tmp = t_0
	elif x <= -4.8e-92:
		tmp = t_1
	elif x <= -6.2e-154:
		tmp = x - (z * y)
	elif x <= 3.3e-109:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	t_1 = Float64(z * Float64(-sin(y)))
	tmp = 0.0
	if (x <= -1.45e-28)
		tmp = t_0;
	elseif (x <= -4.8e-92)
		tmp = t_1;
	elseif (x <= -6.2e-154)
		tmp = Float64(x - Float64(z * y));
	elseif (x <= 3.3e-109)
		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 = z * -sin(y);
	tmp = 0.0;
	if (x <= -1.45e-28)
		tmp = t_0;
	elseif (x <= -4.8e-92)
		tmp = t_1;
	elseif (x <= -6.2e-154)
		tmp = x - (z * y);
	elseif (x <= 3.3e-109)
		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[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[x, -1.45e-28], t$95$0, If[LessEqual[x, -4.8e-92], t$95$1, If[LessEqual[x, -6.2e-154], N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-109], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45000000000000006e-28 or 3.2999999999999999e-109 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 82.9%

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

   if -1.45000000000000006e-28 < x < -4.8000000000000002e-92 or -6.19999999999999963e-154 < x < 3.2999999999999999e-109

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 76.5%

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

      1. neg-mul-1 76.5%

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

      2. distribute-rgt-neg-in 76.5%

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

   5. Simplified 76.5%

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

   if -4.8000000000000002e-92 < x < -6.19999999999999963e-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.5%

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

      1. mul-1-neg 84.5%

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

      2. unsub-neg 84.5%

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

   5. Simplified 84.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-92}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-154}:\\ \;\;\;\;x - z \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-109}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00075 \lor \neg \left(y \leq 33000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + \left(z \cdot y\right) \cdot 0.16666666666666666\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00075) (not (<= y 33000.0)))
   (* x (cos y))
   (+ x (* y (- (* y (+ (* x -0.5) (* (* z y) 0.16666666666666666))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00075) || !(y <= 33000.0)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - 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 ((y <= (-0.00075d0)) .or. (.not. (y <= 33000.0d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * ((y * ((x * (-0.5d0)) + ((z * y) * 0.16666666666666666d0))) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00075) || !(y <= 33000.0)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00075) or not (y <= 33000.0):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00075) || !(y <= 33000.0))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(Float64(z * y) * 0.16666666666666666))) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00075) || ~((y <= 33000.0)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((y * ((x * -0.5) + ((z * y) * 0.16666666666666666))) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00075], N[Not[LessEqual[y, 33000.0]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000002e-4 or 33000 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 53.9%

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

   if -7.5000000000000002e-4 < y < 33000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00075 \lor \neg \left(y \leq 33000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + \left(z \cdot y\right) \cdot 0.16666666666666666\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+212} \lor \neg \left(z \leq 3.9 \cdot 10^{+119}\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.5e+212) (not (<= z 3.9e+119))) (* z (- y)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e+212) || !(z <= 3.9e+119)) {
		tmp = z * -y;
	} 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 <= (-2.5d+212)) .or. (.not. (z <= 3.9d+119))) then
        tmp = z * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.5e+212) || !(z <= 3.9e+119)) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.5e+212) or not (z <= 3.9e+119):
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.5e+212) || !(z <= 3.9e+119))
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.5e+212) || ~((z <= 3.9e+119)))
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.5e+212], N[Not[LessEqual[z, 3.9e+119]], $MachinePrecision]], N[(z * (-y)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999996e212 or 3.8999999999999998e119 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 50.7%

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

      1. mul-1-neg 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around 0 41.7%

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

      1. associate-*r* 41.7%

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

      2. neg-mul-1 41.7%

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

      3. *-commutative 41.7%

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

   8. Simplified 41.7%

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

   if -2.49999999999999996e212 < z < 3.8999999999999998e119

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 51.7%

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

      1. mul-1-neg 51.7%

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

      2. unsub-neg 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around inf 44.2%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+212} \lor \neg \left(z \leq 3.9 \cdot 10^{+119}\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.1% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in y around 0 51.5%

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

   1. mul-1-neg 51.5%

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

   2. unsub-neg 51.5%

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

5. Simplified 51.5%

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

6. Final simplification 51.5%

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

Alternative 10: 38.5% 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 99.7%

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

3. Taylor expanded in y around 0 51.5%

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

   1. mul-1-neg 51.5%

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

   2. unsub-neg 51.5%

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

5. Simplified 51.5%

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

6. Taylor expanded in x around inf 37.7%

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

Reproduce

herbie shell --seed 2024103 
(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))))