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

Percentage Accurate: 99.8% → 99.8%

Time: 13.2s

Alternatives: 7

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.04 \cdot 10^{+105}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;y \leq -0.00195 \lor \neg \left(y \leq 0.0095\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right) + \left(x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -9.2e+254)
     t_0
     (if (<= y -1.04e+105)
       (* z (- (sin y)))
       (if (or (<= y -0.00195) (not (<= y 0.0095)))
         t_0
         (+ (* -0.5 (* x (* y y))) (- x (* y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -9.2e+254) {
		tmp = t_0;
	} else if (y <= -1.04e+105) {
		tmp = z * -sin(y);
	} else if ((y <= -0.00195) || !(y <= 0.0095)) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (x * (y * y))) + (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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (y <= (-9.2d+254)) then
        tmp = t_0
    else if (y <= (-1.04d+105)) then
        tmp = z * -sin(y)
    else if ((y <= (-0.00195d0)) .or. (.not. (y <= 0.0095d0))) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * (x * (y * y))) + (x - (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 (y <= -9.2e+254) {
		tmp = t_0;
	} else if (y <= -1.04e+105) {
		tmp = z * -Math.sin(y);
	} else if ((y <= -0.00195) || !(y <= 0.0095)) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (x * (y * y))) + (x - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if y <= -9.2e+254:
		tmp = t_0
	elif y <= -1.04e+105:
		tmp = z * -math.sin(y)
	elif (y <= -0.00195) or not (y <= 0.0095):
		tmp = t_0
	else:
		tmp = (-0.5 * (x * (y * y))) + (x - (y * z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -9.2e+254)
		tmp = t_0;
	elseif (y <= -1.04e+105)
		tmp = Float64(z * Float64(-sin(y)));
	elseif ((y <= -0.00195) || !(y <= 0.0095))
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * Float64(x * Float64(y * y))) + Float64(x - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (y <= -9.2e+254)
		tmp = t_0;
	elseif (y <= -1.04e+105)
		tmp = z * -sin(y);
	elseif ((y <= -0.00195) || ~((y <= 0.0095)))
		tmp = t_0;
	else
		tmp = (-0.5 * (x * (y * y))) + (x - (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[y, -9.2e+254], t$95$0, If[LessEqual[y, -1.04e+105], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[y, -0.00195], N[Not[LessEqual[y, 0.0095]], $MachinePrecision]], t$95$0, N[(N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.19999999999999994e254 or -1.04e105 < y < -0.0019499999999999999 or 0.00949999999999999976 < y

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 40.1%

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

      2. pow2 40.1%

         \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \cos y - z \cdot \sin y}\right)}^{2}} \]

   3. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \cos y - z \cdot \sin y}\right)}^{2}} \]

   4. Taylor expanded in z around 0 60.4%

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

   if -9.19999999999999994e254 < y < -1.04e105

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 65.9%

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

      1. mul-1-neg 65.9%

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

      2. *-commutative 65.9%

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

      3. distribute-rgt-neg-in 65.9%

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

   4. Simplified 65.9%

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

   if -0.0019499999999999999 < y < 0.00949999999999999976

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. fma-def 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. associate--l+ 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+254}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq -1.04 \cdot 10^{+105}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;y \leq -0.00195 \lor \neg \left(y \leq 0.0095\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right) + \left(x - y \cdot z\right)\\ \end{array} \]

Alternative 3: 85.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.88 \cdot 10^{-34} \lor \neg \left(z \leq 1.85 \cdot 10^{-134}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.88e-34) (not (<= z 1.85e-134)))
   (- x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.88e-34) || !(z <= 1.85e-134)) {
		tmp = x - (z * sin(y));
	} 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 <= (-1.88d-34)) .or. (.not. (z <= 1.85d-134))) then
        tmp = x - (z * sin(y))
    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 <= -1.88e-34) || !(z <= 1.85e-134)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.88e-34) or not (z <= 1.85e-134):
		tmp = x - (z * math.sin(y))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.88e-34) || !(z <= 1.85e-134))
		tmp = Float64(x - Float64(z * sin(y)));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.88e-34) || ~((z <= 1.85e-134)))
		tmp = x - (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.88e-34], N[Not[LessEqual[z, 1.85e-134]], $MachinePrecision]], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.88e-34 or 1.85e-134 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 87.0%

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

   if -1.88e-34 < z < 1.85e-134

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 47.2%

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

      2. pow2 47.2%

         \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \cos y - z \cdot \sin y}\right)}^{2}} \]

   3. Applied egg-rr 47.2%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \cos y - z \cdot \sin y}\right)}^{2}} \]

   4. Taylor expanded in z around 0 93.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.88 \cdot 10^{-34} \lor \neg \left(z \leq 1.85 \cdot 10^{-134}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00195 \lor \neg \left(y \leq 0.0082\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right) + \left(x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00195) (not (<= y 0.0082)))
   (* x (cos y))
   (+ (* -0.5 (* x (* y y))) (- x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00195) || !(y <= 0.0082)) {
		tmp = x * cos(y);
	} else {
		tmp = (-0.5 * (x * (y * y))) + (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 <= (-0.00195d0)) .or. (.not. (y <= 0.0082d0))) then
        tmp = x * cos(y)
    else
        tmp = ((-0.5d0) * (x * (y * y))) + (x - (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00195) || !(y <= 0.0082)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = (-0.5 * (x * (y * y))) + (x - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00195) or not (y <= 0.0082):
		tmp = x * math.cos(y)
	else:
		tmp = (-0.5 * (x * (y * y))) + (x - (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00195) || !(y <= 0.0082))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(Float64(-0.5 * Float64(x * Float64(y * y))) + Float64(x - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00195) || ~((y <= 0.0082)))
		tmp = x * cos(y);
	else
		tmp = (-0.5 * (x * (y * y))) + (x - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00195], N[Not[LessEqual[y, 0.0082]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0019499999999999999 or 0.00820000000000000069 < y

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 41.0%

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

      2. pow2 41.0%

         \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \cos y - z \cdot \sin y}\right)}^{2}} \]

   3. Applied egg-rr 41.0%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \cos y - z \cdot \sin y}\right)}^{2}} \]

   4. Taylor expanded in z around 0 54.2%

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

   if -0.0019499999999999999 < y < 0.00820000000000000069

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. fma-def 99.6%

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

      5. unpow2 99.6%

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

      6. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. associate--l+ 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00195 \lor \neg \left(y \leq 0.0082\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right) + \left(x - y \cdot z\right)\\ \end{array} \]

Alternative 5: 41.7% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+245} \lor \neg \left(z \leq 4.6 \cdot 10^{+149}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9e+245) (not (<= z 4.6e+149))) (* y (- z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e+245) || !(z <= 4.6e+149)) {
		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 <= (-9d+245)) .or. (.not. (z <= 4.6d+149))) 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 <= -9e+245) || !(z <= 4.6e+149)) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9e+245) or not (z <= 4.6e+149):
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9e+245) || !(z <= 4.6e+149))
		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 <= -9e+245) || ~((z <= 4.6e+149)))
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9e+245], N[Not[LessEqual[z, 4.6e+149]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -9e245 or 4.5999999999999997e149 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 54.9%

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

      1. +-commutative 54.9%

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

      2. mul-1-neg 54.9%

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

      3. unsub-neg 54.9%

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

      4. fma-def 54.9%

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

      5. unpow2 54.9%

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

      6. associate-*l* 54.9%

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

   4. Simplified 54.9%

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

      1. fma-udef 54.9%

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

      2. associate--l+ 54.9%

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

      3. associate-*r* 54.9%

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

      4. *-commutative 54.9%

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

      5. *-commutative 54.9%

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

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in x around 0 42.9%

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

      1. mul-1-neg 42.9%

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

      2. distribute-rgt-neg-in 42.9%

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

   9. Simplified 42.9%

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

   if -9e245 < z < 4.5999999999999997e149

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 52.0%

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

      1. +-commutative 52.0%

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

      2. mul-1-neg 52.0%

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

      3. unsub-neg 52.0%

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

      4. fma-def 52.0%

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

      5. unpow2 52.0%

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

      6. associate-*l* 52.0%

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

   4. Simplified 52.0%

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

   5. Taylor expanded in y around 0 46.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+245} \lor \neg \left(z \leq 4.6 \cdot 10^{+149}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 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. Taylor expanded in y around 0 53.8%

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

   1. +-commutative 53.8%

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

   2. mul-1-neg 53.8%

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

   3. unsub-neg 53.8%

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

4. Simplified 53.8%

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

5. Final simplification 53.8%

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

Alternative 7: 38.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0 52.4%

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

   1. +-commutative 52.4%

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

   2. mul-1-neg 52.4%

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

   3. unsub-neg 52.4%

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

   4. fma-def 52.4%

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

   5. unpow2 52.4%

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

   6. associate-*l* 52.5%

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

4. Simplified 52.5%

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

5. Taylor expanded in y around 0 40.9%

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

6. Final simplification 40.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023279 
(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))))