Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 10.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. cos-lowering-cos.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 74.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ \mathbf{if}\;y \leq -375:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+200}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)))
   (if (<= y -375.0)
     t_0
     (if (<= y 3.4e-21) (fma z y x) (if (<= y 4.8e+200) (* x (cos y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double tmp;
	if (y <= -375.0) {
		tmp = t_0;
	} else if (y <= 3.4e-21) {
		tmp = fma(z, y, x);
	} else if (y <= 4.8e+200) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	tmp = 0.0
	if (y <= -375.0)
		tmp = t_0;
	elseif (y <= 3.4e-21)
		tmp = fma(z, y, x);
	elseif (y <= 4.8e+200)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -375.0], t$95$0, If[LessEqual[y, 3.4e-21], N[(z * y + x), $MachinePrecision], If[LessEqual[y, 4.8e+200], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -375 or 4.8000000000000001e200 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 58.2

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

   5. Simplified 58.2%

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

   if -375 < y < 3.4e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.4

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

   5. Simplified 99.4%

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

   if 3.4e-21 < y < 4.8000000000000001e200

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. cos-lowering-cos.f64 69.8

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

   5. Simplified 69.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -375:\\ \;\;\;\;\sin y \cdot z\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+200}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) z x)))
   (if (<= z -1.1e-45) t_0 (if (<= z 1.1e-127) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), z, x);
	double tmp;
	if (z <= -1.1e-45) {
		tmp = t_0;
	} else if (z <= 1.1e-127) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), z, x)
	tmp = 0.0
	if (z <= -1.1e-45)
		tmp = t_0;
	elseif (z <= 1.1e-127)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1.1e-45], t$95$0, If[LessEqual[z, 1.1e-127], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.09999999999999997e-45 or 1.1000000000000001e-127 < z

   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 82.7%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. sin-lowering-sin.f64 82.7

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

   7. Applied egg-rr 82.7%

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

   if -1.09999999999999997e-45 < z < 1.1000000000000001e-127

   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. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 96.2

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

   5. Simplified 96.2%

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

Alternative 4: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0044:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0044) t_0 (if (<= y 3.4e-21) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.0044) {
		tmp = t_0;
	} else if (y <= 3.4e-21) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.0044)
		tmp = t_0;
	elseif (y <= 3.4e-21)
		tmp = fma(z, y, 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, -0.0044], t$95$0, If[LessEqual[y, 3.4e-21], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00440000000000000027 or 3.4e-21 < 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. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 55.7

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

   5. Simplified 55.7%

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

   if -0.00440000000000000027 < y < 3.4e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 5: 40.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z -1.85e+134) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.85e+134) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.85d+134)) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.85e+134) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.85e+134:
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.85e+134)
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.85e+134)
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.85e+134], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.85000000000000007e134

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 46.4

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

   5. Simplified 46.4%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 37.9

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

   8. Simplified 37.9%

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

   if -1.85000000000000007e134 < z

   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 44.7%

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

Alternative 6: 52.8% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 52.1

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

5. Simplified 52.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
6. Add Preprocessing

Alternative 7: 39.3% 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 40.3%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))