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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

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 99.9%

   \[x \cdot \cos y - z \cdot \sin y \]
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. Final simplification 99.9%

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

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.00095:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.215:\\ \;\;\;\;x - y \cdot z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))) (t_1 (* x (cos y))))
   (if (<= y -1.9e+56)
     t_0
     (if (<= y -0.00095)
       t_1
       (if (<= y 0.215) (- x (* y z)) (if (<= y 2.3e+40) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -1.9e+56) {
		tmp = t_0;
	} else if (y <= -0.00095) {
		tmp = t_1;
	} else if (y <= 0.215) {
		tmp = x - (y * z);
	} else if (y <= 2.3e+40) {
		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 = sin(y) * -z
    t_1 = x * cos(y)
    if (y <= (-1.9d+56)) then
        tmp = t_0
    else if (y <= (-0.00095d0)) then
        tmp = t_1
    else if (y <= 0.215d0) then
        tmp = x - (y * z)
    else if (y <= 2.3d+40) 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 = Math.sin(y) * -z;
	double t_1 = x * Math.cos(y);
	double tmp;
	if (y <= -1.9e+56) {
		tmp = t_0;
	} else if (y <= -0.00095) {
		tmp = t_1;
	} else if (y <= 0.215) {
		tmp = x - (y * z);
	} else if (y <= 2.3e+40) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	t_1 = x * math.cos(y)
	tmp = 0
	if y <= -1.9e+56:
		tmp = t_0
	elif y <= -0.00095:
		tmp = t_1
	elif y <= 0.215:
		tmp = x - (y * z)
	elif y <= 2.3e+40:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.9e+56)
		tmp = t_0;
	elseif (y <= -0.00095)
		tmp = t_1;
	elseif (y <= 0.215)
		tmp = Float64(x - Float64(y * z));
	elseif (y <= 2.3e+40)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	t_1 = x * cos(y);
	tmp = 0.0;
	if (y <= -1.9e+56)
		tmp = t_0;
	elseif (y <= -0.00095)
		tmp = t_1;
	elseif (y <= 0.215)
		tmp = x - (y * z);
	elseif (y <= 2.3e+40)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.9e+56], t$95$0, If[LessEqual[y, -0.00095], t$95$1, If[LessEqual[y, 0.215], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+40], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.89999999999999998e56 or 2.29999999999999994e40 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 67.4%

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

      1. associate-*r* 67.4%

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

      2. neg-mul-1 67.4%

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

   4. Simplified 67.4%

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

   if -1.89999999999999998e56 < y < -9.49999999999999998e-4 or 0.214999999999999997 < y < 2.29999999999999994e40

   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 76.1%

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

   if -9.49999999999999998e-4 < y < 0.214999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+56}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -0.00095:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.215:\\ \;\;\;\;x - y \cdot z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]

Alternative 4: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.0011:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.215:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))) (t_1 (* x (cos y))))
   (if (<= y -1.6e+56)
     t_0
     (if (<= y -0.0011)
       t_1
       (if (<= y 0.215) (fma (- y) z x) (if (<= y 4.25e+43) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -1.6e+56) {
		tmp = t_0;
	} else if (y <= -0.0011) {
		tmp = t_1;
	} else if (y <= 0.215) {
		tmp = fma(-y, z, x);
	} else if (y <= 4.25e+43) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.6e+56)
		tmp = t_0;
	elseif (y <= -0.0011)
		tmp = t_1;
	elseif (y <= 0.215)
		tmp = fma(Float64(-y), z, x);
	elseif (y <= 4.25e+43)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+56], t$95$0, If[LessEqual[y, -0.0011], t$95$1, If[LessEqual[y, 0.215], N[((-y) * z + x), $MachinePrecision], If[LessEqual[y, 4.25e+43], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000002e56 or 4.25e43 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 67.4%

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

      1. associate-*r* 67.4%

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

      2. neg-mul-1 67.4%

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

   4. Simplified 67.4%

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

   if -1.60000000000000002e56 < y < -0.00110000000000000007 or 0.214999999999999997 < y < 4.25e43

   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 76.1%

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

   if -0.00110000000000000007 < y < 0.214999999999999997

   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 99.5%

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

      1. associate-*r* 99.5%

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

      2. neg-mul-1 99.5%

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

      3. fma-def 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+56}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq -0.0011:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.215:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]

Alternative 5: 83.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+192}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.15e+127)
     t_0
     (if (<= x 8.8e+192) (- x (* (sin y) z)) (- t_0 (* y z))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.15e+127) {
		tmp = t_0;
	} else if (x <= 8.8e+192) {
		tmp = x - (sin(y) * z);
	} else {
		tmp = t_0 - (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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (x <= (-1.15d+127)) then
        tmp = t_0
    else if (x <= 8.8d+192) then
        tmp = x - (sin(y) * z)
    else
        tmp = t_0 - (y * z)
    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 tmp;
	if (x <= -1.15e+127) {
		tmp = t_0;
	} else if (x <= 8.8e+192) {
		tmp = x - (Math.sin(y) * z);
	} else {
		tmp = t_0 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.15e+127:
		tmp = t_0
	elif x <= 8.8e+192:
		tmp = x - (math.sin(y) * z)
	else:
		tmp = t_0 - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.15e+127)
		tmp = t_0;
	elseif (x <= 8.8e+192)
		tmp = Float64(x - Float64(sin(y) * z));
	else
		tmp = Float64(t_0 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -1.15e+127)
		tmp = t_0;
	elseif (x <= 8.8e+192)
		tmp = x - (sin(y) * z);
	else
		tmp = t_0 - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+127], t$95$0, If[LessEqual[x, 8.8e+192], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1500000000000001e127

   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 z around 0 89.6%

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

   if -1.1500000000000001e127 < x < 8.8000000000000003e192

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 86.2%

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

   if 8.8000000000000003e192 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+192}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y - y \cdot z\\ \end{array} \]

Alternative 6: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+128} \lor \neg \left(x \leq 6.5 \cdot 10^{+192}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.8e+128) (not (<= x 6.5e+192)))
   (* x (cos y))
   (- x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.8e+128) || !(x <= 6.5e+192)) {
		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 <= (-4.8d+128)) .or. (.not. (x <= 6.5d+192))) 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 <= -4.8e+128) || !(x <= 6.5e+192)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.8e+128) or not (x <= 6.5e+192):
		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 <= -4.8e+128) || !(x <= 6.5e+192))
		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 <= -4.8e+128) || ~((x <= 6.5e+192)))
		tmp = x * cos(y);
	else
		tmp = x - (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.8e+128], N[Not[LessEqual[x, 6.5e+192]], $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 < -4.8000000000000004e128 or 6.50000000000000033e192 < 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 z around 0 92.1%

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

   if -4.8000000000000004e128 < x < 6.50000000000000033e192

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 86.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+128} \lor \neg \left(x \leq 6.5 \cdot 10^{+192}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]

Alternative 7: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0021 \lor \neg \left(y \leq 0.215\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0021) (not (<= y 0.215))) (* x (cos y)) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0021) || !(y <= 0.215)) {
		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 <= (-0.0021d0)) .or. (.not. (y <= 0.215d0))) 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 <= -0.0021) || !(y <= 0.215)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0021) or not (y <= 0.215):
		tmp = x * math.cos(y)
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0021) || !(y <= 0.215))
		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 <= -0.0021) || ~((y <= 0.215)))
		tmp = x * cos(y);
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0021], N[Not[LessEqual[y, 0.215]], $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 < -0.00209999999999999987 or 0.214999999999999997 < y

   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 41.0%

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

   if -0.00209999999999999987 < y < 0.214999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0021 \lor \neg \left(y \leq 0.215\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]

Alternative 8: 42.2% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+166} \lor \neg \left(z \leq 7 \cdot 10^{+147}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.9e+166) (not (<= z 7e+147))) (* y (- z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e+166) || !(z <= 7e+147)) {
		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 ((z <= (-3.9d+166)) .or. (.not. (z <= 7d+147))) 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 ((z <= -3.9e+166) || !(z <= 7e+147)) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.9e+166) or not (z <= 7e+147):
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.9e+166) || !(z <= 7e+147))
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.9e+166) || ~((z <= 7e+147)))
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.9e+166], N[Not[LessEqual[z, 7e+147]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.89999999999999991e166 or 6.99999999999999949e147 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. mul-1-neg 43.8%

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

      3. unsub-neg 43.8%

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

   4. Simplified 43.8%

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

   5. Taylor expanded in x around 0 31.4%

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

      1. associate-*r* 31.4%

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

      2. neg-mul-1 31.4%

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

   7. Simplified 31.4%

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

   if -3.89999999999999991e166 < z < 6.99999999999999949e147

   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 52.3%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+166} \lor \neg \left(z \leq 7 \cdot 10^{+147}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 52.3% 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 99.9%

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

2. Taylor expanded in y around 0 55.4%

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

   1. +-commutative 55.4%

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

   2. mul-1-neg 55.4%

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

   3. unsub-neg 55.4%

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

4. Simplified 55.4%

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

5. Final simplification 55.4%

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

Alternative 10: 38.9% 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 99.9%

   \[x \cdot \cos y - z \cdot \sin y \]
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 43.1%

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

5. Final simplification 43.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023200 
(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))))