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

Percentage Accurate: 99.8% → 99.8%

Time: 11.5s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z * (-N[Sin[y], $MachinePrecision]) + 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. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-def 99.8%

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

   7. sin-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 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 3: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-131}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-67}:\\ \;\;\;\;t_0 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.9e-28)
     t_0
     (if (<= x 2.25e-131)
       (* z (- (sin y)))
       (if (<= x 2.35e-67) (- t_0 (* z y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.9e-28) {
		tmp = t_0;
	} else if (x <= 2.25e-131) {
		tmp = z * -sin(y);
	} else if (x <= 2.35e-67) {
		tmp = t_0 - (z * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(y)
    if (x <= (-1.9d-28)) then
        tmp = t_0
    else if (x <= 2.25d-131) then
        tmp = z * -sin(y)
    else if (x <= 2.35d-67) then
        tmp = t_0 - (z * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.cos(y);
	double tmp;
	if (x <= -1.9e-28) {
		tmp = t_0;
	} else if (x <= 2.25e-131) {
		tmp = z * -Math.sin(y);
	} else if (x <= 2.35e-67) {
		tmp = t_0 - (z * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.9e-28:
		tmp = t_0
	elif x <= 2.25e-131:
		tmp = z * -math.sin(y)
	elif x <= 2.35e-67:
		tmp = t_0 - (z * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.9e-28)
		tmp = t_0;
	elseif (x <= 2.25e-131)
		tmp = Float64(z * Float64(-sin(y)));
	elseif (x <= 2.35e-67)
		tmp = Float64(t_0 - Float64(z * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -1.9e-28)
		tmp = t_0;
	elseif (x <= 2.25e-131)
		tmp = z * -sin(y);
	elseif (x <= 2.35e-67)
		tmp = t_0 - (z * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e-28], t$95$0, If[LessEqual[x, 2.25e-131], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 2.35e-67], N[(t$95$0 - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.90000000000000005e-28 or 2.35000000000000002e-67 < x

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-def 99.8%

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

      7. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 84.9%

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

   if -1.90000000000000005e-28 < x < 2.2500000000000001e-131

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-def 99.8%

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

      7. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 79.0%

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

      1. neg-mul-1 79.0%

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

      2. *-commutative 79.0%

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

      3. distribute-rgt-neg-in 79.0%

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

   6. Simplified 79.0%

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

   if 2.2500000000000001e-131 < x < 2.35000000000000002e-67

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 88.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-131}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \cos y - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-28} \lor \neg \left(x \leq 7.2 \cdot 10^{-130}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.3e-28) (not (<= x 7.2e-130)))
   (* x (cos y))
   (* z (- (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.3e-28) || !(x <= 7.2e-130)) {
		tmp = x * cos(y);
	} else {
		tmp = z * -sin(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 ((x <= (-2.3d-28)) .or. (.not. (x <= 7.2d-130))) then
        tmp = x * cos(y)
    else
        tmp = z * -sin(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.3e-28) || !(x <= 7.2e-130)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = z * -Math.sin(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.3e-28) or not (x <= 7.2e-130):
		tmp = x * math.cos(y)
	else:
		tmp = z * -math.sin(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.3e-28) || !(x <= 7.2e-130))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(z * Float64(-sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.3e-28) || ~((x <= 7.2e-130)))
		tmp = x * cos(y);
	else
		tmp = z * -sin(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.3e-28], N[Not[LessEqual[x, 7.2e-130]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999986e-28 or 7.2000000000000003e-130 < x

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-def 99.8%

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

      7. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 82.6%

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

   if -2.29999999999999986e-28 < x < 7.2000000000000003e-130

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. sin-neg 99.8%

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

      6. fma-def 99.8%

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

      7. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 79.0%

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

      1. neg-mul-1 79.0%

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

      2. *-commutative 79.0%

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

      3. distribute-rgt-neg-in 79.0%

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

   6. Simplified 79.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-28} \lor \neg \left(x \leq 7.2 \cdot 10^{-130}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]

Alternative 5: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.038 \lor \neg \left(y \leq 0.00145\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.038) (not (<= y 0.00145)))
   (* x (cos y))
   (+ x (* y (- (* y (* x -0.5)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.038) || !(y <= 0.00145)) {
		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 <= (-0.038d0)) .or. (.not. (y <= 0.00145d0))) 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 <= -0.038) || !(y <= 0.00145)) {
		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 <= -0.038) or not (y <= 0.00145):
		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 <= -0.038) || !(y <= 0.00145))
		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 <= -0.038) || ~((y <= 0.00145)))
		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, -0.038], N[Not[LessEqual[y, 0.00145]], $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 < -0.0379999999999999991 or 0.00145 < y

   1. Initial program 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-lft-neg-out 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. sin-neg 99.5%

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

      6. fma-def 99.6%

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

      7. sin-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 51.6%

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

   if -0.0379999999999999991 < y < 0.00145

   1. Initial program 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-out 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. sin-neg 100.0%

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

      6. fma-def 100.0%

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

      7. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

      4. associate-*r* 99.0%

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

      5. unpow2 99.0%

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

      6. associate-*r* 99.0%

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

      7. *-commutative 99.0%

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

      8. distribute-rgt-out-- 99.0%

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

      9. *-commutative 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.038 \lor \neg \left(y \leq 0.00145\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5\right) - z\right)\\ \end{array} \]

Alternative 6: 53.2% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-def 99.8%

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

   7. sin-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 53.5%

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

   1. mul-1-neg 53.5%

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

   2. unsub-neg 53.5%

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

6. Simplified 53.5%

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

7. Final simplification 53.5%

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

Alternative 7: 17.1% accurate, 51.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. cancel-sign-sub-inv 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-neg-out 99.8%

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

   4. distribute-rgt-neg-in 99.8%

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

   5. sin-neg 99.8%

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

   6. fma-def 99.8%

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

   7. sin-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 38.8%

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

   1. neg-mul-1 38.8%

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

   2. *-commutative 38.8%

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

   3. distribute-rgt-neg-in 38.8%

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

6. Simplified 38.8%

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

7. Taylor expanded in y around 0 15.6%

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

   1. mul-1-neg 15.6%

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

   2. distribute-rgt-neg-in 15.6%

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

9. Simplified 15.6%

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

10. Final simplification 15.6%

    \[\leadsto -z \cdot y \]

Reproduce

herbie shell --seed 2023275 
(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))))