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

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- 0.0 (sin y)) z (* x (cos y))))

C

double code(double x, double y, double z) {
	return fma((0.0 - sin(y)), z, (x * cos(y)));
}

Julia

function code(x, y, z)
	return fma(Float64(0.0 - sin(y)), z, Float64(x * cos(y)))
end

Wolfram

code[x_, y_, z_] := N[(N[(0.0 - N[Sin[y], $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\sin y}, z, x \cdot \cos y\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(0 - \sin y, z, \color{blue}{x \cdot \cos y}\right) \]

   10. cos-lowering-cos.f64 99.8

       \[\leadsto \mathsf{fma}\left(0 - \sin y, z, x \cdot \color{blue}{\cos y}\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sin-lowering-sin.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right) \]
8. Add Preprocessing

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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 3: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+111}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -2.75e+104) t_0 (if (<= x 1.12e+111) (- x (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -2.75e+104) {
		tmp = t_0;
	} else if (x <= 1.12e+111) {
		tmp = 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 = x * cos(y)
    if (x <= (-2.75d+104)) then
        tmp = t_0
    else if (x <= 1.12d+111) then
        tmp = 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 = x * Math.cos(y);
	double tmp;
	if (x <= -2.75e+104) {
		tmp = t_0;
	} else if (x <= 1.12e+111) {
		tmp = x - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -2.75e+104:
		tmp = t_0
	elif x <= 1.12e+111:
		tmp = x - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -2.75e+104)
		tmp = t_0;
	elseif (x <= 1.12e+111)
		tmp = Float64(x - Float64(sin(y) * z));
	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 <= -2.75e+104)
		tmp = t_0;
	elseif (x <= 1.12e+111)
		tmp = x - (sin(y) * z);
	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, -2.75e+104], t$95$0, If[LessEqual[x, 1.12e+111], N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75000000000000008e104 or 1.11999999999999995e111 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \cos y + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, 0\right)} \]

      3. cos-lowering-cos.f64 93.2

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\cos y}, 0\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 93.2

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

   7. Applied egg-rr 93.2%

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

   if -2.75000000000000008e104 < x < 1.11999999999999995e111

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 86.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+111}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-77}:\\ \;\;\;\;0 - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.1e-48) t_0 (if (<= x 1.9e-77) (- 0.0 (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.1e-48) {
		tmp = t_0;
	} else if (x <= 1.9e-77) {
		tmp = 0.0 - (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 = x * cos(y)
    if (x <= (-1.1d-48)) then
        tmp = t_0
    else if (x <= 1.9d-77) then
        tmp = 0.0d0 - (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 = x * Math.cos(y);
	double tmp;
	if (x <= -1.1e-48) {
		tmp = t_0;
	} else if (x <= 1.9e-77) {
		tmp = 0.0 - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.1e-48:
		tmp = t_0
	elif x <= 1.9e-77:
		tmp = 0.0 - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.1e-48)
		tmp = t_0;
	elseif (x <= 1.9e-77)
		tmp = Float64(0.0 - Float64(sin(y) * z));
	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 <= -1.1e-48)
		tmp = t_0;
	elseif (x <= 1.9e-77)
		tmp = 0.0 - (sin(y) * z);
	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, -1.1e-48], t$95$0, If[LessEqual[x, 1.9e-77], N[(0.0 - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000006e-48 or 1.8999999999999999e-77 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \cos y + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, 0\right)} \]

      3. cos-lowering-cos.f64 83.1

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\cos y}, 0\right) \]

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 83.1

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

   7. Applied egg-rr 83.1%

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

   if -1.10000000000000006e-48 < x < 1.8999999999999999e-77

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\sin y}, z, x \cdot \cos y\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(0 - \sin y, z, \color{blue}{x \cdot \cos y}\right) \]

      10. cos-lowering-cos.f64 99.7

          \[\leadsto \mathsf{fma}\left(0 - \sin y, z, x \cdot \color{blue}{\cos y}\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - z \cdot \sin y} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - z \cdot \sin y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{z \cdot \sin y} \]

      5. sin-lowering-sin.f64 73.9

         \[\leadsto 0 - z \cdot \color{blue}{\sin y} \]

   7. Simplified 73.9%

      \[\leadsto \color{blue}{0 - z \cdot \sin y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-77}:\\ \;\;\;\;0 - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.024:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, -0.5, z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.024)
     t_0
     (if (<= y 0.026)
       (fma y (- (* y (fma x -0.5 (* z (* y 0.16666666666666666)))) z) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.024) {
		tmp = t_0;
	} else if (y <= 0.026) {
		tmp = fma(y, ((y * fma(x, -0.5, (z * (y * 0.16666666666666666)))) - z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.024)
		tmp = t_0;
	elseif (y <= 0.026)
		tmp = fma(y, Float64(Float64(y * fma(x, -0.5, Float64(z * Float64(y * 0.16666666666666666)))) - z), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.024], t$95$0, If[LessEqual[y, 0.026], N[(y * N[(N[(y * N[(x * -0.5 + N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.024 or 0.0259999999999999988 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \cos y + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, 0\right)} \]

      3. cos-lowering-cos.f64 52.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\cos y}, 0\right) \]

   5. Simplified 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 52.8

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

   7. Applied egg-rr 52.8%

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

   if -0.024 < y < 0.0259999999999999988

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\sin y}, z, x \cdot \cos y\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(0 - \sin y, z, \color{blue}{x \cdot \cos y}\right) \]

      10. cos-lowering-cos.f64 100.0

          \[\leadsto \mathsf{fma}\left(0 - \sin y, z, x \cdot \color{blue}{\cos y}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. sin-lowering-sin.f64 100.0

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{0 - \sin y \cdot z}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{0 - \sin y \cdot z}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, 0 - \color{blue}{\sin y \cdot z}\right) \]

      9. sin-lowering-sin.f64 100.0

         \[\leadsto \mathsf{fma}\left(\cos y, x, 0 - \color{blue}{\sin y} \cdot z\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, x, 0 - \sin y \cdot z\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{6} \cdot \left(y \cdot z\right)\right) - z\right)} \]
   10. 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. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.024:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(x, -0.5, z \cdot \left(y \cdot 0.16666666666666666\right)\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{+135}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4.6e+135) x (- 0.0 (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.6e+135) {
		tmp = x;
	} else {
		tmp = 0.0 - (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 <= 4.6d+135) then
        tmp = x
    else
        tmp = 0.0d0 - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.6e+135) {
		tmp = x;
	} else {
		tmp = 0.0 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.6e+135:
		tmp = x
	else:
		tmp = 0.0 - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.6e+135)
		tmp = x;
	else
		tmp = Float64(0.0 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 4.6e+135)
		tmp = x;
	else
		tmp = 0.0 - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.6e+135], x, N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.6000000000000002e135

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 46.0%

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

   if 4.6000000000000002e135 < z

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\sin y}, z, x \cdot \cos y\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(0 - \sin y, z, \color{blue}{x \cdot \cos y}\right) \]

      10. cos-lowering-cos.f64 99.8

          \[\leadsto \mathsf{fma}\left(0 - \sin y, z, x \cdot \color{blue}{\cos y}\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - z}, x\right) \]

      6. --lowering--.f64 53.1

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - z}, x\right) \]

   7. Simplified 53.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0 - z, x\right)} \]

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - y \cdot z} \]

      4. *-lowering-*.f64 39.7

         \[\leadsto 0 - \color{blue}{y \cdot z} \]

   10. Simplified 39.7%

       \[\leadsto \color{blue}{0 - y \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.4% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 0 - z, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(y, Float64(0.0 - z), x)
end

Wolfram

code[x_, y_, z_] := N[(y * N[(0.0 - z), $MachinePrecision] + 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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \sin y}, z, x \cdot \cos y\right) \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\sin y}, z, x \cdot \cos y\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(0 - \sin y, z, \color{blue}{x \cdot \cos y}\right) \]

   10. cos-lowering-cos.f64 99.8

       \[\leadsto \mathsf{fma}\left(0 - \sin y, z, x \cdot \color{blue}{\cos y}\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - \sin y, z, x \cdot \cos y\right)} \]

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - z}, x\right) \]

   6. --lowering--.f64 51.9

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 - z}, x\right) \]

7. Simplified 51.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0 - z, x\right)} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 51.9

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

9. Applied egg-rr 51.9%

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

10. Final simplification 51.9%

    \[\leadsto \mathsf{fma}\left(y, 0 - z, x\right) \]
11. Add Preprocessing

Alternative 8: 39.6% accurate, 214.0× 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. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 41.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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))))