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

Percentage Accurate: 99.8% → 99.8%

Time: 7.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 85.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000000 \lor \neg \left(x \leq 3.2 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10000000.0) (not (<= x 3.2e-59)))
   (* x (cos y))
   (- x (* (sin y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10000000.0) || !(x <= 3.2e-59)) {
		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 <= (-10000000.0d0)) .or. (.not. (x <= 3.2d-59))) 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 <= -10000000.0) || !(x <= 3.2e-59)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x - (Math.sin(y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -10000000.0) or not (x <= 3.2e-59):
		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 <= -10000000.0) || !(x <= 3.2e-59))
		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 <= -10000000.0) || ~((x <= 3.2e-59)))
		tmp = x * cos(y);
	else
		tmp = x - (sin(y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -10000000.0], N[Not[LessEqual[x, 3.2e-59]], $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 < -1e7 or 3.1999999999999999e-59 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 87.0%

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

   if -1e7 < x < 3.1999999999999999e-59

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 90.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000 \lor \neg \left(x \leq 3.2 \cdot 10^{-59}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]

Alternative 4: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-33} \lor \neg \left(x \leq 1.3 \cdot 10^{-122}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.8e-33) (not (<= x 1.3e-122)))
   (* x (cos y))
   (* (sin y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.8e-33) || !(x <= 1.3e-122)) {
		tmp = x * cos(y);
	} 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 <= (-5.8d-33)) .or. (.not. (x <= 1.3d-122))) then
        tmp = x * cos(y)
    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 <= -5.8e-33) || !(x <= 1.3e-122)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = Math.sin(y) * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.8e-33) or not (x <= 1.3e-122):
		tmp = x * math.cos(y)
	else:
		tmp = math.sin(y) * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.8e-33) || !(x <= 1.3e-122))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(sin(y) * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.8e-33) || ~((x <= 1.3e-122)))
		tmp = x * cos(y);
	else
		tmp = sin(y) * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.8e-33], N[Not[LessEqual[x, 1.3e-122]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.80000000000000005e-33 or 1.29999999999999988e-122 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around 0 84.0%

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

   if -5.80000000000000005e-33 < x < 1.29999999999999988e-122

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 70.4%

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

      1. mul-1-neg 70.4%

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

      2. *-commutative 70.4%

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

      3. distribute-rgt-neg-in 70.4%

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

   4. Simplified 70.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-33} \lor \neg \left(x \leq 1.3 \cdot 10^{-122}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]

Alternative 5: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1400000 \lor \neg \left(y \leq 0.013\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1400000.0) (not (<= y 0.013)))
   (* x (cos y))
   (+ x (* y (- (* y (* x -0.5)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1400000.0) || !(y <= 0.013)) {
		tmp = x * cos(y);
	} 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 <= (-1400000.0d0)) .or. (.not. (y <= 0.013d0))) then
        tmp = x * cos(y)
    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 <= -1400000.0) || !(y <= 0.013)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((y * (x * -0.5)) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1400000.0) or not (y <= 0.013):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((y * (x * -0.5)) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1400000.0) || !(y <= 0.013))
		tmp = Float64(x * cos(y));
	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 <= -1400000.0) || ~((y <= 0.013)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((y * (x * -0.5)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1400000.0], N[Not[LessEqual[y, 0.013]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $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 < -1.4e6 or 0.0129999999999999994 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 53.7%

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

   if -1.4e6 < y < 0.0129999999999999994

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 53.3%

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

      3. associate-*r* 53.3%

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

      4. fma-neg 53.3%

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

      5. distribute-rgt-neg-in 53.3%

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

   3. Applied egg-rr 53.3%

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

   4. Taylor expanded in y around 0 53.3%

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

      1. mul-1-neg 53.3%

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

      2. distribute-rgt-neg-in 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. +-commutative 98.2%

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

      3. associate-+l+ 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*r* 98.2%

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

      6. metadata-eval 98.2%

         \[\leadsto x + \left({y}^{2} \cdot \left(x \cdot \color{blue}{\left(-0.3333333333333333 + -0.16666666666666666\right)}\right) + -1 \cdot \left(y \cdot z\right)\right) \]

      7. distribute-rgt-out 98.2%

         \[\leadsto x + \left({y}^{2} \cdot \color{blue}{\left(-0.3333333333333333 \cdot x + -0.16666666666666666 \cdot x\right)} + -1 \cdot \left(y \cdot z\right)\right) \]

      8. mul-1-neg 98.2%

         \[\leadsto x + \left({y}^{2} \cdot \left(-0.3333333333333333 \cdot x + -0.16666666666666666 \cdot x\right) + \color{blue}{\left(-y \cdot z\right)}\right) \]

      9. unsub-neg 98.2%

         \[\leadsto x + \color{blue}{\left({y}^{2} \cdot \left(-0.3333333333333333 \cdot x + -0.16666666666666666 \cdot x\right) - y \cdot z\right)} \]

      10. unpow2 98.2%

          \[\leadsto x + \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(-0.3333333333333333 \cdot x + -0.16666666666666666 \cdot x\right) - y \cdot z\right) \]

      11. associate-*l* 98.2%

          \[\leadsto x + \left(\color{blue}{y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot x + -0.16666666666666666 \cdot x\right)\right)} - y \cdot z\right) \]

      12. distribute-lft-out-- 98.2%

          \[\leadsto x + \color{blue}{y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot x + -0.16666666666666666 \cdot x\right) - z\right)} \]

      13. distribute-rgt-out 98.2%

          \[\leadsto x + y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(-0.3333333333333333 + -0.16666666666666666\right)\right)} - z\right) \]

      14. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1400000 \lor \neg \left(y \leq 0.013\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5\right) - z\right)\\ \end{array} \]

Alternative 6: 40.3% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+215}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.4e+215) x (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+215) {
		tmp = x;
	} else {
		tmp = 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 (z <= 1.4d+215) then
        tmp = x
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+215) {
		tmp = x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.4e+215:
		tmp = x
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.4e+215)
		tmp = x;
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.4e+215)
		tmp = x;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.4e+215], x, N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e215

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 42.1%

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

   if 1.4e215 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. fma-def 64.4%

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

      5. unpow2 64.4%

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

      6. associate-*l* 65.0%

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

   4. Simplified 65.0%

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

   5. Taylor expanded in x around 0 53.6%

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

      1. neg-mul-1 53.6%

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

      2. distribute-lft-neg-in 53.6%

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

   7. Simplified 53.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+215}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 7: 52.8% 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 50.5%

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

   1. +-commutative 50.5%

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

   2. mul-1-neg 50.5%

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

   3. unsub-neg 50.5%

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

4. Simplified 50.5%

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

5. Final simplification 50.5%

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

Alternative 8: 39.1% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around 0 40.1%

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

5. Final simplification 40.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023258 
(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))))