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

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-149}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-79} \lor \neg \left(x \leq 1.95 \cdot 10^{+23}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -5.5e-84)
     t_0
     (if (<= x 6.4e-149)
       (* z (- (sin y)))
       (if (or (<= x 3.4e-79) (not (<= x 1.95e+23))) t_0 (- x (* y z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -5.5e-84) {
		tmp = t_0;
	} else if (x <= 6.4e-149) {
		tmp = z * -sin(y);
	} else if ((x <= 3.4e-79) || !(x <= 1.95e+23)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (x <= (-5.5d-84)) then
        tmp = t_0
    else if (x <= 6.4d-149) then
        tmp = z * -sin(y)
    else if ((x <= 3.4d-79) .or. (.not. (x <= 1.95d+23))) then
        tmp = t_0
    else
        tmp = 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 (x <= -5.5e-84) {
		tmp = t_0;
	} else if (x <= 6.4e-149) {
		tmp = z * -Math.sin(y);
	} else if ((x <= 3.4e-79) || !(x <= 1.95e+23)) {
		tmp = t_0;
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -5.5e-84:
		tmp = t_0
	elif x <= 6.4e-149:
		tmp = z * -math.sin(y)
	elif (x <= 3.4e-79) or not (x <= 1.95e+23):
		tmp = t_0
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -5.5e-84)
		tmp = t_0;
	elseif (x <= 6.4e-149)
		tmp = Float64(z * Float64(-sin(y)));
	elseif ((x <= 3.4e-79) || !(x <= 1.95e+23))
		tmp = t_0;
	else
		tmp = 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 (x <= -5.5e-84)
		tmp = t_0;
	elseif (x <= 6.4e-149)
		tmp = z * -sin(y);
	elseif ((x <= 3.4e-79) || ~((x <= 1.95e+23)))
		tmp = t_0;
	else
		tmp = 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[x, -5.5e-84], t$95$0, If[LessEqual[x, 6.4e-149], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[x, 3.4e-79], N[Not[LessEqual[x, 1.95e+23]], $MachinePrecision]], t$95$0, N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.50000000000000019e-84 or 6.40000000000000004e-149 < x < 3.39999999999999976e-79 or 1.95e23 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 85.2%

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

   if -5.50000000000000019e-84 < x < 6.40000000000000004e-149

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. neg-mul-1 76.3%

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

   4. Simplified 76.3%

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

   if 3.39999999999999976e-79 < x < 1.95e23

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. unsub-neg 76.8%

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

   4. Simplified 76.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-149}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-79} \lor \neg \left(x \leq 1.95 \cdot 10^{+23}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-141} \lor \neg \left(z \leq 1.8 \cdot 10^{-80}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.2e-141) (not (<= z 1.8e-80)))
   (- x (* z (sin y)))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e-141) || !(z <= 1.8e-80)) {
		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 <= (-2.2d-141)) .or. (.not. (z <= 1.8d-80))) 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 <= -2.2e-141) || !(z <= 1.8e-80)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.2e-141) or not (z <= 1.8e-80):
		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 <= -2.2e-141) || !(z <= 1.8e-80))
		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 <= -2.2e-141) || ~((z <= 1.8e-80)))
		tmp = x - (z * sin(y));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.2e-141], N[Not[LessEqual[z, 1.8e-80]], $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 < -2.20000000000000009e-141 or 1.8e-80 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 90.7%

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

   if -2.20000000000000009e-141 < z < 1.8e-80

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 86.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-141} \lor \neg \left(z \leq 1.8 \cdot 10^{-80}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00082 \lor \neg \left(y \leq 0.034\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot \left(\left(y \cdot y\right) \cdot -0.5\right) - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00082) (not (<= y 0.034)))
   (* x (cos y))
   (+ x (- (* x (* (* y y) -0.5)) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00082) || !(y <= 0.034)) {
		tmp = x * cos(y);
	} else {
		tmp = x + ((x * ((y * y) * -0.5)) - (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 <= (-0.00082d0)) .or. (.not. (y <= 0.034d0))) then
        tmp = x * cos(y)
    else
        tmp = x + ((x * ((y * y) * (-0.5d0))) - (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00082) || !(y <= 0.034)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + ((x * ((y * y) * -0.5)) - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00082) or not (y <= 0.034):
		tmp = x * math.cos(y)
	else:
		tmp = x + ((x * ((y * y) * -0.5)) - (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00082) || !(y <= 0.034))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(Float64(x * Float64(Float64(y * y) * -0.5)) - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00082) || ~((y <= 0.034)))
		tmp = x * cos(y);
	else
		tmp = x + ((x * ((y * y) * -0.5)) - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00082], N[Not[LessEqual[y, 0.034]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x * N[(N[(y * y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.1999999999999998e-4 or 0.034000000000000002 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 48.8%

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

   if -8.1999999999999998e-4 < y < 0.034000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. unpow2 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00082 \lor \neg \left(y \leq 0.034\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot \left(\left(y \cdot y\right) \cdot -0.5\right) - y \cdot z\right)\\ \end{array} \]

Alternative 5: 41.0% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-275}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-178) x (if (<= x 1.1e-275) (* z (- y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-178) {
		tmp = x;
	} else if (x <= 1.1e-275) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.3d-178)) then
        tmp = x
    else if (x <= 1.1d-275) then
        tmp = z * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-178) {
		tmp = x;
	} else if (x <= 1.1e-275) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.3e-178:
		tmp = x
	elif x <= 1.1e-275:
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-178)
		tmp = x;
	elseif (x <= 1.1e-275)
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e-178)
		tmp = x;
	elseif (x <= 1.1e-275)
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e-178], x, If[LessEqual[x, 1.1e-275], N[(z * (-y)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999994e-178 or 1.09999999999999994e-275 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 56.9%

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

      1. mul-1-neg 56.9%

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

      2. unsub-neg 56.9%

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

   4. Simplified 56.9%

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

   5. Taylor expanded in x around inf 47.9%

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

   if -2.29999999999999994e-178 < x < 1.09999999999999994e-275

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.3%

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

      1. mul-1-neg 65.3%

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

      2. unsub-neg 65.3%

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

   4. Simplified 65.3%

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

      1. sub-neg 65.3%

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

      2. flip-+ 27.8%

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

      3. *-commutative 27.8%

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

      4. distribute-rgt-neg-in 27.8%

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

      5. *-commutative 27.8%

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

      6. distribute-rgt-neg-in 27.8%

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

      7. *-commutative 27.8%

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

      8. distribute-rgt-neg-in 27.8%

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

   6. Applied egg-rr 27.8%

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

   7. Taylor expanded in x around 0 53.1%

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

      1. mul-1-neg 53.1%

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

      2. *-commutative 53.1%

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

      3. distribute-rgt-neg-in 53.1%

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

   9. Simplified 53.1%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-275}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 52.6% 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 58.2%

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

   1. mul-1-neg 58.2%

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

   2. unsub-neg 58.2%

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

4. Simplified 58.2%

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

5. Final simplification 58.2%

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

Alternative 7: 2.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 99.9%

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

   1. prod-diff 99.9%

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

   2. *-commutative 99.9%

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

   3. fma-neg 99.9%

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

   4. add-cube-cbrt 97.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y} \cdot \sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right) \cdot \sqrt[3]{x \cdot \cos y - z \cdot \sin y}} + \mathsf{fma}\left(-\sin y, z, \sin y \cdot z\right) \]

   5. fma-def 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y} \cdot \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \mathsf{fma}\left(-\sin y, z, \sin y \cdot z\right)\right)} \]

   6. pow2 97.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \mathsf{fma}\left(-\sin y, z, \sin y \cdot z\right)\right) \]

   7. *-commutative 97.9%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \mathsf{fma}\left(-\sin y, z, \color{blue}{z \cdot \sin y}\right)\right) \]

   8. fma-udef 97.9%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \color{blue}{\left(-\sin y\right) \cdot z + z \cdot \sin y}\right) \]

   9. distribute-lft-neg-in 97.9%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \color{blue}{\left(-\sin y \cdot z\right)} + z \cdot \sin y\right) \]

   10. *-commutative 97.9%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \left(-\color{blue}{z \cdot \sin y}\right) + z \cdot \sin y\right) \]

   11. distribute-rgt-neg-in 97.9%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \color{blue}{z \cdot \left(-\sin y\right)} + z \cdot \sin y\right) \]

   12. fma-def 97.9%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \color{blue}{\mathsf{fma}\left(z, -\sin y, z \cdot \sin y\right)}\right) \]

3. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x \cdot \cos y - z \cdot \sin y}\right)}^{2}, \sqrt[3]{x \cdot \cos y - z \cdot \sin y}, \mathsf{fma}\left(z, -\sin y, z \cdot \sin y\right)\right)} \]

4. Taylor expanded in x around inf 3.1%

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

   1. distribute-lft1-in 3.1%

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

   2. metadata-eval 3.1%

      \[\leadsto \color{blue}{0} \cdot \left(z \cdot \sin y\right) \]

   3. mul0-lft 3.1%

      \[\leadsto \color{blue}{0} \]

6. Simplified 3.1%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 3.1%

   \[\leadsto 0 \]

Alternative 8: 38.7% 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. Taylor expanded in y around 0 58.2%

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

   1. mul-1-neg 58.2%

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

   2. unsub-neg 58.2%

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

4. Simplified 58.2%

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

5. Taylor expanded in x around inf 42.7%

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

6. Final simplification 42.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(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))))