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

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-rgt-neg-out 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-udef 99.8%

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

   5. distribute-rgt-neg-out 99.8%

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

   6. *-commutative 99.8%

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

   7. distribute-rgt-neg-in 99.8%

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

4. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 86.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+90} \lor \neg \left(x \leq 1.2 \cdot 10^{+25}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.1e+90) (not (<= x 1.2e+25)))
   (* (cos y) x)
   (- x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.1e+90) || !(x <= 1.2e+25)) {
		tmp = cos(y) * x;
	} 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 <= (-4.1d+90)) .or. (.not. (x <= 1.2d+25))) then
        tmp = cos(y) * x
    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 <= -4.1e+90) || !(x <= 1.2e+25)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = x - (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.1e+90) or not (x <= 1.2e+25):
		tmp = math.cos(y) * x
	else:
		tmp = x - (math.sin(y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.1e+90) || !(x <= 1.2e+25))
		tmp = Float64(cos(y) * x);
	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 <= -4.1e+90) || ~((x <= 1.2e+25)))
		tmp = cos(y) * x;
	else
		tmp = x - (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.1e+90], N[Not[LessEqual[x, 1.2e+25]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000042e90 or 1.19999999999999998e25 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-udef 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. *-commutative 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around inf 85.6%

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

   if -4.10000000000000042e90 < x < 1.19999999999999998e25

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 88.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+90} \lor \neg \left(x \leq 1.2 \cdot 10^{+25}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]

Alternative 4: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-56} \lor \neg \left(x \leq 6.5 \cdot 10^{-136}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.5e-56) (not (<= x 6.5e-136)))
   (* (cos y) x)
   (* (sin y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e-56) || !(x <= 6.5e-136)) {
		tmp = cos(y) * x;
	} else {
		tmp = 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 <= (-9.5d-56)) .or. (.not. (x <= 6.5d-136))) then
        tmp = cos(y) * x
    else
        tmp = sin(y) * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e-56) || !(x <= 6.5e-136)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = Math.sin(y) * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.5e-56) or not (x <= 6.5e-136):
		tmp = math.cos(y) * x
	else:
		tmp = math.sin(y) * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.5e-56) || !(x <= 6.5e-136))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(sin(y) * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.5e-56) || ~((x <= 6.5e-136)))
		tmp = cos(y) * x;
	else
		tmp = sin(y) * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.5e-56], N[Not[LessEqual[x, 6.5e-136]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999991e-56 or 6.50000000000000011e-136 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-udef 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. *-commutative 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around inf 77.3%

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

   if -9.4999999999999991e-56 < x < 6.50000000000000011e-136

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 81.8%

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

      1. mul-1-neg 81.8%

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

      2. *-commutative 81.8%

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

      3. distribute-rgt-neg-in 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-56} \lor \neg \left(x \leq 6.5 \cdot 10^{-136}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.05 \lor \neg \left(y \leq 5200000000\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.05) (not (<= y 5200000000.0)))
   (* (cos y) x)
   (+ x (* y (- (* y (* x -0.5)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.05) || !(y <= 5200000000.0)) {
		tmp = cos(y) * x;
	} else {
		tmp = x + (y * ((y * (x * -0.5)) - 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 <= (-0.05d0)) .or. (.not. (y <= 5200000000.0d0))) then
        tmp = cos(y) * x
    else
        tmp = x + (y * ((y * (x * (-0.5d0))) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.05) || !(y <= 5200000000.0)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = x + (y * ((y * (x * -0.5)) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.05) or not (y <= 5200000000.0):
		tmp = math.cos(y) * x
	else:
		tmp = x + (y * ((y * (x * -0.5)) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.05) || !(y <= 5200000000.0))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(x * -0.5)) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.05) || ~((y <= 5200000000.0)))
		tmp = cos(y) * x;
	else
		tmp = x + (y * ((y * (x * -0.5)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.05], N[Not[LessEqual[y, 5200000000.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(x + N[(y * N[(N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.050000000000000003 or 5.2e9 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-out 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-udef 99.5%

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

      5. distribute-rgt-neg-out 99.5%

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

      6. *-commutative 99.5%

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

      7. distribute-rgt-neg-in 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around inf 54.2%

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

   if -0.050000000000000003 < y < 5.2e9

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. distribute-rgt-in 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. flip-+ 66.0%

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

      4. *-commutative 66.0%

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

      5. *-commutative 66.0%

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

      6. *-commutative 66.0%

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

      7. associate-*l* 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*l* 66.0%

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

      10. *-commutative 66.0%

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

      11. *-commutative 66.0%

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

      12. associate-*l* 66.0%

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

   7. Applied egg-rr 66.0%

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

   8. Taylor expanded in y around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

      3. +-commutative 98.1%

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

      4. +-commutative 98.1%

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

      5. associate-*r* 98.1%

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

      6. *-commutative 98.1%

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

      7. unpow2 98.1%

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

      8. associate-*r* 98.1%

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

      9. associate-*r* 98.1%

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

      10. distribute-rgt-neg-out 98.1%

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

      11. sub-neg 98.1%

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

      12. associate--l+ 98.1%

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

      13. distribute-lft-out-- 98.1%

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

      14. associate-*l* 98.1%

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

   10. Simplified 98.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.05 \lor \neg \left(y \leq 5200000000\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5\right) - z\right)\\ \end{array} \]

Alternative 6: 40.5% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e-69) x (if (<= x 2.2e-174) (* y (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e-69) {
		tmp = x;
	} else if (x <= 2.2e-174) {
		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 <= (-1.55d-69)) then
        tmp = x
    else if (x <= 2.2d-174) 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 <= -1.55e-69) {
		tmp = x;
	} else if (x <= 2.2e-174) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55e-69:
		tmp = x
	elif x <= 2.2e-174:
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e-69)
		tmp = x;
	elseif (x <= 2.2e-174)
		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 <= -1.55e-69)
		tmp = x;
	elseif (x <= 2.2e-174)
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e-69], x, If[LessEqual[x, 2.2e-174], N[(y * (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e-69 or 2.20000000000000022e-174 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-udef 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. *-commutative 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 44.5%

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

   if -1.55e-69 < x < 2.20000000000000022e-174

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 78.0%

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

      1. unpow2 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in y around 0 64.5%

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

   6. Taylor expanded in x around 0 53.7%

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

      1. associate-*r* 53.7%

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

      2. neg-mul-1 53.7%

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

   8. Simplified 53.7%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-69}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 52.2% 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 56.1%

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

   1. +-commutative 56.1%

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

   2. mul-1-neg 56.1%

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

   3. unsub-neg 56.1%

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

4. Simplified 56.1%

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

5. Final simplification 56.1%

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

Alternative 8: 38.8% 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 x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-rgt-neg-out 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-udef 99.8%

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

   5. distribute-rgt-neg-out 99.8%

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

   6. *-commutative 99.8%

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

   7. distribute-rgt-neg-in 99.8%

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

4. Simplified 99.8%

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

5. Taylor expanded in y around 0 35.7%

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

6. Final simplification 35.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023182 
(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))))