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

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \cos y \cdot x - \sin y \cdot z \]
4. Add Preprocessing

Alternative 2: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 750000:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= x -3.2e+71)
     t_0
     (if (<= x 750000.0) (- (* 1.0 x) (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (x <= -3.2e+71) {
		tmp = t_0;
	} else if (x <= 750000.0) {
		tmp = (1.0 * x) - (sin(y) * z);
	} 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 = cos(y) * x
    if (x <= (-3.2d+71)) then
        tmp = t_0
    else if (x <= 750000.0d0) then
        tmp = (1.0d0 * x) - (sin(y) * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * x;
	double tmp;
	if (x <= -3.2e+71) {
		tmp = t_0;
	} else if (x <= 750000.0) {
		tmp = (1.0 * x) - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * x
	tmp = 0
	if x <= -3.2e+71:
		tmp = t_0
	elif x <= 750000.0:
		tmp = (1.0 * x) - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (x <= -3.2e+71)
		tmp = t_0;
	elseif (x <= 750000.0)
		tmp = Float64(Float64(1.0 * x) - Float64(sin(y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * x;
	tmp = 0.0;
	if (x <= -3.2e+71)
		tmp = t_0;
	elseif (x <= 750000.0)
		tmp = (1.0 * x) - (sin(y) * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.2e+71], t$95$0, If[LessEqual[x, 750000.0], N[(N[(1.0 * x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000023e71 or 7.5e5 < x

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -3.20000000000000023e71 < x < 7.5e5

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]
   4. Step-by-step derivation

   5. Applied rewrites 86.9%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+71}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;x \leq 750000:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+194}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -3.5e+49) t_0 (if (<= z 2.3e+194) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -3.5e+49) {
		tmp = t_0;
	} else if (z <= 2.3e+194) {
		tmp = cos(y) * x;
	} 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 = -z * sin(y)
    if (z <= (-3.5d+49)) then
        tmp = t_0
    else if (z <= 2.3d+194) then
        tmp = cos(y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z * Math.sin(y);
	double tmp;
	if (z <= -3.5e+49) {
		tmp = t_0;
	} else if (z <= 2.3e+194) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * math.sin(y)
	tmp = 0
	if z <= -3.5e+49:
		tmp = t_0
	elif z <= 2.3e+194:
		tmp = math.cos(y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (z <= -3.5e+49)
		tmp = t_0;
	elseif (z <= 2.3e+194)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * sin(y);
	tmp = 0.0;
	if (z <= -3.5e+49)
		tmp = t_0;
	elseif (z <= 2.3e+194)
		tmp = cos(y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e+49], t$95$0, If[LessEqual[z, 2.3e+194], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.49999999999999975e49 or 2.30000000000000005e194 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -3.49999999999999975e49 < z < 2.30000000000000005e194

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 75.8

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

   5. Applied rewrites 75.8%

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

Alternative 4: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.225:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.12:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.225)
     t_0
     (if (<= y 0.12)
       (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.225) {
		tmp = t_0;
	} else if (y <= 0.12) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.225)
		tmp = t_0;
	elseif (y <= 0.12)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.225], t$95$0, If[LessEqual[y, 0.12], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.225000000000000006 or 0.12 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if -0.225000000000000006 < y < 0.12

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right) \cdot y} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z, y, x\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z}, y, x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y} - z, y, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y} - z, y, x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{-1}{2} \cdot x\right)} \cdot y - z, y, x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot z, \frac{-1}{2} \cdot x\right)} \cdot y - z, y, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{z \cdot y}, \frac{-1}{2} \cdot x\right) \cdot y - z, y, x\right) \]

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, \color{blue}{-0.5 \cdot x}\right) \cdot y - z, y, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+197}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 8e+197) (* 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8e+197) {
		tmp = 1.0 * 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 <= 8d+197) then
        tmp = 1.0d0 * 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 <= 8e+197) {
		tmp = 1.0 * x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 8e+197:
		tmp = 1.0 * x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 8e+197)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 8e+197)
		tmp = 1.0 * x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 8e+197], N[(1.0 * x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.9999999999999996e197

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(x \cdot \cos y\right)}^{3}} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(x \cdot \cos y\right)}}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{{\color{blue}{\left(\cos y \cdot x\right)}}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\cos y \cdot x\right)}}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - \color{blue}{{\left(z \cdot \sin y\right)}^{3}}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\color{blue}{\left(z \cdot \sin y\right)}}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\color{blue}{\left(\sin y \cdot z\right)}}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\color{blue}{\left(\sin y \cdot z\right)}}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\left(\sin y \cdot z\right)}^{3}}{\color{blue}{\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right) + \left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)}} \]

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 36.9%

       \[\leadsto 1 \cdot x \]

   if 7.9999999999999996e197 < z

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 41.7%

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

Alternative 6: 52.3% accurate, 23.8× 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. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 44.1

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

5. Applied rewrites 44.1%

   \[\leadsto \color{blue}{x - z \cdot y} \]
6. Add Preprocessing

Alternative 7: 38.6% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 * x;
}

Python

def code(x, y, z):
	return 1.0 * x

Julia

function code(x, y, z)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(x \cdot \cos y\right)}^{3} - {\left(z \cdot \sin y\right)}^{3}}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(x \cdot \cos y\right)}^{3}} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(x \cdot \cos y\right)}}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{{\color{blue}{\left(\cos y \cdot x\right)}}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(\cos y \cdot x\right)}}^{3} - {\left(z \cdot \sin y\right)}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - \color{blue}{{\left(z \cdot \sin y\right)}^{3}}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\color{blue}{\left(z \cdot \sin y\right)}}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\color{blue}{\left(\sin y \cdot z\right)}}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\color{blue}{\left(\sin y \cdot z\right)}}^{3}}{\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right) + \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{{\left(\cos y \cdot x\right)}^{3} - {\left(\sin y \cdot z\right)}^{3}}{\color{blue}{\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right) + \left(x \cdot \cos y\right) \cdot \left(z \cdot \sin y\right)\right) + \left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)}} \]

4. Applied rewrites 39.6%

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

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 57.8

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

7. Applied rewrites 57.8%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 33.8%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(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))))