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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 9

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 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-def 99.8%

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

   7. sin-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+140} \lor \neg \left(x \leq 6.8 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+140) (not (<= x 6.8e-12)))
   (* x (cos y))
   (fma z (- (sin y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+140) || !(x <= 6.8e-12)) {
		tmp = x * cos(y);
	} else {
		tmp = fma(z, -sin(y), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e+140) || !(x <= 6.8e-12))
		tmp = Float64(x * cos(y));
	else
		tmp = fma(z, Float64(-sin(y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+140], N[Not[LessEqual[x, 6.8e-12]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z * (-N[Sin[y], $MachinePrecision]) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999987e140 or 6.8000000000000001e-12 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around inf 92.4%

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

   if -1.94999999999999987e140 < x < 6.8000000000000001e-12

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-def 99.9%

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

      7. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 90.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+140} \lor \neg \left(x \leq 6.8 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right)\\ \end{array} \]

Alternative 3: 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. Final simplification 99.8%

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

Alternative 4: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.18:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 30500:\\ \;\;\;\;\left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - z \cdot y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+209} \lor \neg \left(y \leq 3.6 \cdot 10^{+235}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (sin y)))) (t_1 (* x (cos y))))
   (if (<= y -3.5e+200)
     t_0
     (if (<= y -0.18)
       t_1
       (if (<= y 30500.0)
         (- (+ x (* -0.5 (* x (* y y)))) (* z y))
         (if (or (<= y 3.1e+209) (not (<= y 3.6e+235))) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -sin(y);
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -3.5e+200) {
		tmp = t_0;
	} else if (y <= -0.18) {
		tmp = t_1;
	} else if (y <= 30500.0) {
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y);
	} else if ((y <= 3.1e+209) || !(y <= 3.6e+235)) {
		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 = z * -sin(y)
    t_1 = x * cos(y)
    if (y <= (-3.5d+200)) then
        tmp = t_0
    else if (y <= (-0.18d0)) then
        tmp = t_1
    else if (y <= 30500.0d0) then
        tmp = (x + ((-0.5d0) * (x * (y * y)))) - (z * y)
    else if ((y <= 3.1d+209) .or. (.not. (y <= 3.6d+235))) 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 = z * -Math.sin(y);
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -3.5e+200) {
		tmp = t_0;
	} else if (y <= -0.18) {
		tmp = t_1;
	} else if (y <= 30500.0) {
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y);
	} else if ((y <= 3.1e+209) || !(y <= 3.6e+235)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -math.sin(y)
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -3.5e+200:
		tmp = t_0
	elif y <= -0.18:
		tmp = t_1
	elif y <= 30500.0:
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y)
	elif (y <= 3.1e+209) or not (y <= 3.6e+235):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-sin(y)))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -3.5e+200)
		tmp = t_0;
	elseif (y <= -0.18)
		tmp = t_1;
	elseif (y <= 30500.0)
		tmp = Float64(Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))) - Float64(z * y));
	elseif ((y <= 3.1e+209) || !(y <= 3.6e+235))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -sin(y);
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -3.5e+200)
		tmp = t_0;
	elseif (y <= -0.18)
		tmp = t_1;
	elseif (y <= 30500.0)
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y);
	elseif ((y <= 3.1e+209) || ~((y <= 3.6e+235)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e+200], t$95$0, If[LessEqual[y, -0.18], t$95$1, If[LessEqual[y, 30500.0], N[(N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.1e+209], N[Not[LessEqual[y, 3.6e+235]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.50000000000000006e200 or 3.1000000000000001e209 < y < 3.59999999999999985e235

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 67.9%

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

      1. neg-mul-1 67.9%

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

      2. *-commutative 67.9%

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

      3. distribute-rgt-neg-in 67.9%

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

   4. Simplified 67.9%

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

   if -3.50000000000000006e200 < y < -0.17999999999999999 or 30500 < y < 3.1000000000000001e209 or 3.59999999999999985e235 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around inf 66.1%

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

   if -0.17999999999999999 < y < 30500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. +-commutative 97.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right) + -1 \cdot \left(y \cdot z\right)} \]

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. fma-def 97.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {y}^{2} \cdot x, x\right)} - y \cdot z \]

      5. *-commutative 97.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot {y}^{2}}, x\right) - y \cdot z \]

      6. unpow2 97.7%

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

      7. associate-*r* 97.7%

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

   4. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \left(x \cdot y\right) \cdot y, x\right) - y \cdot z} \]
   5. Step-by-step derivation

      1. fma-udef 97.7%

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

      2. associate-*l* 97.7%

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

   6. Applied egg-rr 97.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+200}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;y \leq -0.18:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 30500:\\ \;\;\;\;\left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - z \cdot y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+209} \lor \neg \left(y \leq 3.6 \cdot 10^{+235}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]

Alternative 5: 85.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+139} \lor \neg \left(x \leq 6.8 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.36e+139) (not (<= x 6.8e-12)))
   (* x (cos y))
   (- x (* z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.36e+139) || !(x <= 6.8e-12)) {
		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 <= (-1.36d+139)) .or. (.not. (x <= 6.8d-12))) 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 <= -1.36e+139) || !(x <= 6.8e-12)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (z * Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.36e+139) or not (x <= 6.8e-12):
		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 <= -1.36e+139) || !(x <= 6.8e-12))
		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 <= -1.36e+139) || ~((x <= 6.8e-12)))
		tmp = x * cos(y);
	else
		tmp = x - (z * sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.36e+139], N[Not[LessEqual[x, 6.8e-12]], $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 < -1.36000000000000004e139 or 6.8000000000000001e-12 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around inf 92.4%

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

   if -1.36000000000000004e139 < x < 6.8000000000000001e-12

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 90.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36 \cdot 10^{+139} \lor \neg \left(x \leq 6.8 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 6: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.18 \lor \neg \left(y \leq 30500\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.18) (not (<= y 30500.0)))
   (* x (cos y))
   (- (+ x (* -0.5 (* x (* y y)))) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.18) || !(y <= 30500.0)) {
		tmp = x * cos(y);
	} else {
		tmp = (x + (-0.5 * (x * (y * 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 ((y <= (-0.18d0)) .or. (.not. (y <= 30500.0d0))) then
        tmp = x * cos(y)
    else
        tmp = (x + ((-0.5d0) * (x * (y * y)))) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.18) || !(y <= 30500.0)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.18) or not (y <= 30500.0):
		tmp = x * math.cos(y)
	else:
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.18) || !(y <= 30500.0))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(Float64(x + Float64(-0.5 * Float64(x * Float64(y * y)))) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.18) || ~((y <= 30500.0)))
		tmp = x * cos(y);
	else
		tmp = (x + (-0.5 * (x * (y * y)))) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.18], N[Not[LessEqual[y, 30500.0]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.17999999999999999 or 30500 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around inf 54.7%

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

   if -0.17999999999999999 < y < 30500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. +-commutative 97.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right) + -1 \cdot \left(y \cdot z\right)} \]

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. fma-def 97.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {y}^{2} \cdot x, x\right)} - y \cdot z \]

      5. *-commutative 97.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot {y}^{2}}, x\right) - y \cdot z \]

      6. unpow2 97.7%

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

      7. associate-*r* 97.7%

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

   4. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \left(x \cdot y\right) \cdot y, x\right) - y \cdot z} \]
   5. Step-by-step derivation

      1. fma-udef 97.7%

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

      2. associate-*l* 97.7%

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

   6. Applied egg-rr 97.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.18 \lor \neg \left(y \leq 30500\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - z \cdot y\\ \end{array} \]

Alternative 7: 42.2% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-133}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-220}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-133) x (if (<= x 8.5e-220) (* z (- y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-133) {
		tmp = x;
	} else if (x <= 8.5e-220) {
		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 (x <= (-2d-133)) then
        tmp = x
    else if (x <= 8.5d-220) 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 (x <= -2e-133) {
		tmp = x;
	} else if (x <= 8.5e-220) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e-133:
		tmp = x
	elif x <= 8.5e-220:
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-133)
		tmp = x;
	elseif (x <= 8.5e-220)
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e-133)
		tmp = x;
	elseif (x <= 8.5e-220)
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e-133], x, If[LessEqual[x, 8.5e-220], N[(z * (-y)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-133 or 8.4999999999999996e-220 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 48.9%

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

   if -2.0000000000000001e-133 < x < 8.4999999999999996e-220

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 59.2%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right)} \]
   3. Step-by-step derivation

      1. +-commutative 59.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot x\right) + x\right) + -1 \cdot \left(y \cdot z\right)} \]

      2. mul-1-neg 59.2%

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

      3. unsub-neg 59.2%

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

      4. fma-def 59.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {y}^{2} \cdot x, x\right)} - y \cdot z \]

      5. *-commutative 59.2%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot {y}^{2}}, x\right) - y \cdot z \]

      6. unpow2 59.2%

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

      7. associate-*r* 59.5%

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

   4. Simplified 59.5%

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

   5. Taylor expanded in x around 0 44.8%

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

      1. mul-1-neg 44.8%

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

      2. distribute-rgt-neg-out 44.8%

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

   7. Simplified 44.8%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-133}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-220}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 52.4% 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 99.8%

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

2. Taylor expanded in y around 0 58.1%

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

   1. +-commutative 58.1%

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

   2. mul-1-neg 58.1%

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

   3. unsub-neg 58.1%

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

4. Simplified 58.1%

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

5. Final simplification 58.1%

   \[\leadsto x - z \cdot y \]

Alternative 9: 39.6% 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. Taylor expanded in x around 0 99.8%

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

   1. neg-mul-1 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in y around 0 41.9%

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

6. Final simplification 41.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023268 
(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))))