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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] * (-z) + 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. *-commutative 99.8%

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

   4. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3600000000000:\\ \;\;\;\;\left(x - y \cdot z\right) + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+79} \lor \neg \left(y \leq 8 \cdot 10^{+257}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))) (t_1 (* (sin y) (- z))))
   (if (<= y -2.4e+104)
     t_0
     (if (<= y -3.9e-6)
       t_1
       (if (<= y 3600000000000.0)
         (+ (- x (* y z)) (* -0.5 (* x (* y y))))
         (if (or (<= y 3.6e+79) (not (<= y 8e+257))) t_0 t_1))))))

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 <= -2.4e+104) {
		tmp = t_0;
	} else if (y <= -3.9e-6) {
		tmp = t_1;
	} else if (y <= 3600000000000.0) {
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)));
	} else if ((y <= 3.6e+79) || !(y <= 8e+257)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 <= (-2.4d+104)) then
        tmp = t_0
    else if (y <= (-3.9d-6)) then
        tmp = t_1
    else if (y <= 3600000000000.0d0) then
        tmp = (x - (y * z)) + ((-0.5d0) * (x * (y * y)))
    else if ((y <= 3.6d+79) .or. (.not. (y <= 8d+257))) then
        tmp = t_0
    else
        tmp = t_1
    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 <= -2.4e+104) {
		tmp = t_0;
	} else if (y <= -3.9e-6) {
		tmp = t_1;
	} else if (y <= 3600000000000.0) {
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)));
	} else if ((y <= 3.6e+79) || !(y <= 8e+257)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	t_1 = math.sin(y) * -z
	tmp = 0
	if y <= -2.4e+104:
		tmp = t_0
	elif y <= -3.9e-6:
		tmp = t_1
	elif y <= 3600000000000.0:
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)))
	elif (y <= 3.6e+79) or not (y <= 8e+257):
		tmp = t_0
	else:
		tmp = t_1
	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 <= -2.4e+104)
		tmp = t_0;
	elseif (y <= -3.9e-6)
		tmp = t_1;
	elseif (y <= 3600000000000.0)
		tmp = Float64(Float64(x - Float64(y * z)) + Float64(-0.5 * Float64(x * Float64(y * y))));
	elseif ((y <= 3.6e+79) || !(y <= 8e+257))
		tmp = t_0;
	else
		tmp = t_1;
	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 <= -2.4e+104)
		tmp = t_0;
	elseif (y <= -3.9e-6)
		tmp = t_1;
	elseif (y <= 3600000000000.0)
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)));
	elseif ((y <= 3.6e+79) || ~((y <= 8e+257)))
		tmp = t_0;
	else
		tmp = t_1;
	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, -2.4e+104], t$95$0, If[LessEqual[y, -3.9e-6], t$95$1, If[LessEqual[y, 3600000000000.0], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 3.6e+79], N[Not[LessEqual[y, 8e+257]], $MachinePrecision]], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e104 or 3.6e12 < y < 3.5999999999999999e79 or 8.00000000000000024e257 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf 67.2%

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

   if -2.4e104 < y < -3.8999999999999999e-6 or 3.5999999999999999e79 < y < 8.00000000000000024e257

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 63.2%

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

      1. associate-*r* 63.2%

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

      2. neg-mul-1 63.2%

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

      3. *-commutative 63.2%

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

   4. Simplified 63.2%

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

   if -3.8999999999999999e-6 < y < 3.6e12

   1. Initial program 100.0%

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

      1. flip3-- 31.2%

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

      2. clear-num 31.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}} \]

      3. clear-num 31.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}}}} \]

      4. flip3-- 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 98.2%

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

      1. associate-+r+ 98.2%

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

      2. mul-1-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. *-commutative 98.2%

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

      5. unpow2 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 3600000000000:\\ \;\;\;\;\left(x - y \cdot z\right) + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+79} \lor \neg \left(y \leq 8 \cdot 10^{+257}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-12} \lor \neg \left(z \leq 7.3 \cdot 10^{-34}\right):\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.5e-12) (not (<= z 7.3e-34)))
   (- x (* (sin y) z))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.5e-12) || !(z <= 7.3e-34)) {
		tmp = x - (sin(y) * z);
	} else {
		tmp = x * cos(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 ((z <= (-4.5d-12)) .or. (.not. (z <= 7.3d-34))) then
        tmp = x - (sin(y) * z)
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.5e-12) || !(z <= 7.3e-34)) {
		tmp = x - (Math.sin(y) * z);
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.5e-12) or not (z <= 7.3e-34):
		tmp = x - (math.sin(y) * z)
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.5e-12) || !(z <= 7.3e-34))
		tmp = Float64(x - Float64(sin(y) * z));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.5e-12) || ~((z <= 7.3e-34)))
		tmp = x - (sin(y) * z);
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.5e-12], N[Not[LessEqual[z, 7.3e-34]], $MachinePrecision]], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.49999999999999981e-12 or 7.29999999999999996e-34 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 88.7%

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

   if -4.49999999999999981e-12 < z < 7.29999999999999996e-34

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 89.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-12} \lor \neg \left(z \leq 7.3 \cdot 10^{-34}\right):\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 5: 73.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6600000 \lor \neg \left(y \leq 3600000000000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6600000.0) (not (<= y 3600000000000.0)))
   (* x (cos y))
   (+ (- x (* y z)) (* -0.5 (* x (* y y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6600000.0) || !(y <= 3600000000000.0)) {
		tmp = x * cos(y);
	} else {
		tmp = (x - (y * z)) + (-0.5 * (x * (y * 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 <= (-6600000.0d0)) .or. (.not. (y <= 3600000000000.0d0))) then
        tmp = x * cos(y)
    else
        tmp = (x - (y * z)) + ((-0.5d0) * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6600000.0) || !(y <= 3600000000000.0)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6600000.0) or not (y <= 3600000000000.0):
		tmp = x * math.cos(y)
	else:
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6600000.0) || !(y <= 3600000000000.0))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(Float64(x - Float64(y * z)) + Float64(-0.5 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6600000.0) || ~((y <= 3600000000000.0)))
		tmp = x * cos(y);
	else
		tmp = (x - (y * z)) + (-0.5 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6600000.0], N[Not[LessEqual[y, 3600000000000.0]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e6 or 3.6e12 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 54.9%

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

   if -6.6e6 < y < 3.6e12

   1. Initial program 99.9%

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

      1. flip3-- 32.0%

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

      2. clear-num 32.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}} \]

      3. clear-num 32.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}}}} \]

      4. flip3-- 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. associate-+r+ 96.6%

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

      2. mul-1-neg 96.6%

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

      3. sub-neg 96.6%

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

      4. *-commutative 96.6%

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

      5. unpow2 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6600000 \lor \neg \left(y \leq 3600000000000\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 39.6% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+218}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.2e+218) x (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.2e+218) {
		tmp = x;
	} else {
		tmp = 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 (z <= 1.2d+218) then
        tmp = x
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.2e+218) {
		tmp = x;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.2e+218:
		tmp = x
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.2e+218)
		tmp = x;
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.2e+218)
		tmp = x;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.2e+218], x, N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1999999999999999e218

   1. Initial program 99.8%

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

      1. flip3-- 34.6%

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

      2. clear-num 34.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}} \]

      3. clear-num 34.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}}}} \]

      4. flip3-- 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 42.9%

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

   if 1.1999999999999999e218 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 64.6%

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

      1. mul-1-neg 64.6%

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

   4. Simplified 64.6%

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

   5. Taylor expanded in x around 0 54.1%

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

      1. associate-*r* 54.1%

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

      2. mul-1-neg 54.1%

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

      3. *-commutative 54.1%

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

   7. Simplified 54.1%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+218}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 7: 52.4% 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. Step-by-step derivation

   1. flip3-- 32.8%

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

   2. clear-num 32.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}} \]

   3. clear-num 32.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}}}} \]

   4. flip3-- 99.6%

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

3. Applied egg-rr 99.6%

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

4. Taylor expanded in y around 0 53.1%

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

   1. mul-1-neg 53.1%

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

   2. sub-neg 53.1%

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

   3. *-commutative 53.1%

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

6. Simplified 53.1%

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

7. Final simplification 53.1%

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

Alternative 8: 38.4% 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. Step-by-step derivation

   1. flip3-- 32.8%

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

   2. clear-num 32.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}} \]

   3. clear-num 32.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}}}} \]

   4. flip3-- 99.6%

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

3. Applied egg-rr 99.6%

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

4. Taylor expanded in y around 0 40.7%

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

5. Final simplification 40.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023297 
(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))))