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

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

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

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

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. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 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: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -3800000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00039:\\ \;\;\;\;x - y \cdot z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+69} \lor \neg \left(y \leq 2.7 \cdot 10^{+122}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= y -3800000.0)
     t_0
     (if (<= y 0.00039)
       (- x (* y z))
       (if (or (<= y 4.8e+69) (not (<= y 2.7e+122))) t_0 (* x (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (y <= -3800000.0) {
		tmp = t_0;
	} else if (y <= 0.00039) {
		tmp = x - (y * z);
	} else if ((y <= 4.8e+69) || !(y <= 2.7e+122)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sin(y) * -z
    if (y <= (-3800000.0d0)) then
        tmp = t_0
    else if (y <= 0.00039d0) then
        tmp = x - (y * z)
    else if ((y <= 4.8d+69) .or. (.not. (y <= 2.7d+122))) then
        tmp = t_0
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) * -z;
	double tmp;
	if (y <= -3800000.0) {
		tmp = t_0;
	} else if (y <= 0.00039) {
		tmp = x - (y * z);
	} else if ((y <= 4.8e+69) || !(y <= 2.7e+122)) {
		tmp = t_0;
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	tmp = 0
	if y <= -3800000.0:
		tmp = t_0
	elif y <= 0.00039:
		tmp = x - (y * z)
	elif (y <= 4.8e+69) or not (y <= 2.7e+122):
		tmp = t_0
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (y <= -3800000.0)
		tmp = t_0;
	elseif (y <= 0.00039)
		tmp = Float64(x - Float64(y * z));
	elseif ((y <= 4.8e+69) || !(y <= 2.7e+122))
		tmp = t_0;
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	tmp = 0.0;
	if (y <= -3800000.0)
		tmp = t_0;
	elseif (y <= 0.00039)
		tmp = x - (y * z);
	elseif ((y <= 4.8e+69) || ~((y <= 2.7e+122)))
		tmp = t_0;
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[y, -3800000.0], t$95$0, If[LessEqual[y, 0.00039], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 4.8e+69], N[Not[LessEqual[y, 2.7e+122]], $MachinePrecision]], t$95$0, N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e6 or 3.89999999999999993e-4 < y < 4.8000000000000003e69 or 2.6999999999999998e122 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 61.6%

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

      1. mul-1-neg 61.6%

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

      2. *-commutative 61.6%

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

      3. distribute-rgt-neg-in 61.6%

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

   4. Simplified 61.6%

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

   if -3.8e6 < y < 3.89999999999999993e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. unsub-neg 98.2%

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

   4. Simplified 98.2%

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

   if 4.8000000000000003e69 < y < 2.6999999999999998e122

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. fma-def 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in z around 0 85.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3800000:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 0.00039:\\ \;\;\;\;x - y \cdot z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+69} \lor \neg \left(y \leq 2.7 \cdot 10^{+122}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -3800000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00096:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+69} \lor \neg \left(y \leq 1.15 \cdot 10^{+111}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= y -3800000.0)
     t_0
     (if (<= y 0.00096)
       (fma (- y) z x)
       (if (or (<= y 5.2e+69) (not (<= y 1.15e+111))) t_0 (* x (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (y <= -3800000.0) {
		tmp = t_0;
	} else if (y <= 0.00096) {
		tmp = fma(-y, z, x);
	} else if ((y <= 5.2e+69) || !(y <= 1.15e+111)) {
		tmp = t_0;
	} else {
		tmp = x * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (y <= -3800000.0)
		tmp = t_0;
	elseif (y <= 0.00096)
		tmp = fma(Float64(-y), z, x);
	elseif ((y <= 5.2e+69) || !(y <= 1.15e+111))
		tmp = t_0;
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[y, -3800000.0], t$95$0, If[LessEqual[y, 0.00096], N[((-y) * z + x), $MachinePrecision], If[Or[LessEqual[y, 5.2e+69], N[Not[LessEqual[y, 1.15e+111]], $MachinePrecision]], t$95$0, N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e6 or 9.60000000000000024e-4 < y < 5.2000000000000004e69 or 1.15000000000000001e111 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 61.6%

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

      1. mul-1-neg 61.6%

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

      2. *-commutative 61.6%

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

      3. distribute-rgt-neg-in 61.6%

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

   4. Simplified 61.6%

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

   if -3.8e6 < y < 9.60000000000000024e-4

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 98.2%

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

      1. associate-*r* 98.2%

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

      2. neg-mul-1 98.2%

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

      3. fma-def 98.2%

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

   6. Simplified 98.2%

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

   if 5.2000000000000004e69 < y < 1.15000000000000001e111

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. fma-def 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in z around 0 85.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3800000:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 0.00096:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+69} \lor \neg \left(y \leq 1.15 \cdot 10^{+111}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 5: 86.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+50} \lor \neg \left(x \leq 1.4 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e+50) (not (<= x 1.4e+108)))
   (* x (cos y))
   (- x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+50) || !(x <= 1.4e+108)) {
		tmp = x * cos(y);
	} else {
		tmp = x - (sin(y) * 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 ((x <= (-1.95d+50)) .or. (.not. (x <= 1.4d+108))) then
        tmp = x * cos(y)
    else
        tmp = x - (sin(y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e+50) || !(x <= 1.4e+108)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e+50) or not (x <= 1.4e+108):
		tmp = x * math.cos(y)
	else:
		tmp = x - (math.sin(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e+50) || !(x <= 1.4e+108))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(sin(y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.95e+50) || ~((x <= 1.4e+108)))
		tmp = x * cos(y);
	else
		tmp = x - (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e+50], N[Not[LessEqual[x, 1.4e+108]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999984e50 or 1.3999999999999999e108 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 84.0%

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

   if -1.94999999999999984e50 < x < 1.3999999999999999e108

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 89.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+50} \lor \neg \left(x \leq 1.4 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]

Alternative 6: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0036 \lor \neg \left(y \leq 0.00044\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0036) (not (<= y 0.00044))) (* x (cos y)) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0036) || !(y <= 0.00044)) {
		tmp = x * cos(y);
	} else {
		tmp = x - (y * 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 <= (-0.0036d0)) .or. (.not. (y <= 0.00044d0))) then
        tmp = x * cos(y)
    else
        tmp = x - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0036) || !(y <= 0.00044)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0036) or not (y <= 0.00044):
		tmp = x * math.cos(y)
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0036) || !(y <= 0.00044))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0036) || ~((y <= 0.00044)))
		tmp = x * cos(y);
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0036], N[Not[LessEqual[y, 0.00044]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0035999999999999999 or 4.40000000000000016e-4 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 44.8%

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

   if -0.0035999999999999999 < y < 4.40000000000000016e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0036 \lor \neg \left(y \leq 0.00044\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]

Alternative 7: 40.9% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.15e-63) x (if (<= x 6.6e-116) (* y (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.15e-63) {
		tmp = x;
	} else if (x <= 6.6e-116) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.15d-63)) then
        tmp = x
    else if (x <= 6.6d-116) then
        tmp = y * -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.15e-63) {
		tmp = x;
	} else if (x <= 6.6e-116) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.15e-63:
		tmp = x
	elif x <= 6.6e-116:
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.15e-63)
		tmp = x;
	elseif (x <= 6.6e-116)
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.15e-63)
		tmp = x;
	elseif (x <= 6.6e-116)
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.15e-63], x, If[LessEqual[x, 6.6e-116], N[(y * (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1499999999999999e-63 or 6.60000000000000002e-116 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 51.3%

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

   if -2.1499999999999999e-63 < x < 6.60000000000000002e-116

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 51.6%

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

      1. +-commutative 51.6%

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

      2. mul-1-neg 51.6%

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

      3. unsub-neg 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in x around 0 44.0%

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

      1. associate-*r* 44.0%

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

      2. neg-mul-1 44.0%

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

   7. Simplified 44.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 40.5% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-122}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.95e-63) (+ x (* y z)) (if (<= x 3e-122) (* y (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e-63) {
		tmp = x + (y * z);
	} else if (x <= 3e-122) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.95d-63)) then
        tmp = x + (y * z)
    else if (x <= 3d-122) then
        tmp = y * -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e-63) {
		tmp = x + (y * z);
	} else if (x <= 3e-122) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.95e-63:
		tmp = x + (y * z)
	elif x <= 3e-122:
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.95e-63)
		tmp = Float64(x + Float64(y * z));
	elseif (x <= 3e-122)
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.95e-63)
		tmp = x + (y * z);
	elseif (x <= 3e-122)
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.95e-63], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-122], N[(y * (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95000000000000011e-63

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 46.2%

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

      1. +-commutative 46.2%

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

      2. mul-1-neg 46.2%

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

      3. unsub-neg 46.2%

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

   4. Simplified 46.2%

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

      1. sub-neg 46.2%

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

      2. +-commutative 46.2%

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

      3. distribute-rgt-neg-in 46.2%

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

      4. add-sqr-sqrt 22.2%

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

      5. sqrt-unprod 34.5%

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

      6. sqr-neg 34.5%

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

      7. sqrt-unprod 22.4%

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

      8. add-sqr-sqrt 42.0%

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

   6. Applied egg-rr 42.0%

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

   if -1.95000000000000011e-63 < x < 3.00000000000000004e-122

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 51.6%

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

      1. +-commutative 51.6%

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

      2. mul-1-neg 51.6%

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

      3. unsub-neg 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in x around 0 44.0%

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

      1. associate-*r* 44.0%

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

      2. neg-mul-1 44.0%

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

   7. Simplified 44.0%

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

   if 3.00000000000000004e-122 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 60.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-122}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 52.1% 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. Taylor expanded in y around 0 55.9%

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

   1. +-commutative 55.9%

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

   2. mul-1-neg 55.9%

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

   3. unsub-neg 55.9%

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

4. Simplified 55.9%

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

5. Final simplification 55.9%

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

Alternative 10: 38.2% 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. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 39.1%

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

5. Final simplification 39.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023244 
(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))))