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

Percentage Accurate: 99.8% → 99.8%

Time: 13.2s

Alternatives: 9

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * x + 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. Step-by-step derivation

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. sin-lowering-sin.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin y}{x}, z \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -2.75e+104)
     t_0
     (if (<= x 2.3e+60) (fma (/ (sin y) x) (* z x) x) 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 <= 2.3e+60) {
		tmp = fma((sin(y) / x), (z * x), x);
	} 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 <= 2.3e+60)
		tmp = fma(Float64(sin(y) / x), Float64(z * x), 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[x, -2.75e+104], t$95$0, If[LessEqual[x, 2.3e+60], N[(N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] * N[(z * x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75000000000000008e104 or 2.30000000000000017e60 < x

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

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

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

      2. cos-lowering-cos.f64 92.4

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

   7. Simplified 92.4%

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

   if -2.75000000000000008e104 < x < 2.30000000000000017e60

   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. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

      7. cos-lowering-cos.f64 89.6

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

   7. Simplified 89.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 90.4

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

   9. Applied egg-rr 90.4%

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

   10. Taylor expanded in y around 0

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

   12. Simplified 77.6%

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

Alternative 4: 74.8% accurate, 1.8× speedup

Math

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

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) (* (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 = 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 = 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 = 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 = 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(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 = 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[(N[Sin[y], $MachinePrecision] * z), $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 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

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

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

      2. cos-lowering-cos.f64 83.3

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

   7. Simplified 83.3%

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

   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. +-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 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf

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

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

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

      2. sin-lowering-sin.f64 74.0

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

   7. Simplified 74.0%

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

4. Final simplification 80.1%

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

Alternative 5: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0015:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00375:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot z\right) \cdot -0.16666666666666666, z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0015)
     t_0
     (if (<= y 0.00375)
       (fma y (fma y (* (* y z) -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 <= -0.0015) {
		tmp = t_0;
	} else if (y <= 0.00375) {
		tmp = fma(y, fma(y, ((y * z) * -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 <= -0.0015)
		tmp = t_0;
	elseif (y <= 0.00375)
		tmp = fma(y, fma(y, Float64(Float64(y * z) * -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, -0.0015], t$95$0, If[LessEqual[y, 0.00375], N[(y * N[(y * N[(N[(y * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0015 or 0.0037499999999999999 < y

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. cos-lowering-cos.f64 53.0

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

   7. Simplified 53.0%

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

   if -0.0015 < y < 0.0037499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

Alternative 6: 51.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot z\right) \cdot -0.16666666666666666, z\right), x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma y (fma y (* (* y z) -0.16666666666666666) z) x))

C

double code(double x, double y, double z) {
	return fma(y, fma(y, ((y * z) * -0.16666666666666666), z), x);
}

Julia

function code(x, y, z)
	return fma(y, fma(y, Float64(Float64(y * z) * -0.16666666666666666), z), x)
end

Wolfram

code[x_, y_, z_] := N[(y * N[(y * N[(N[(y * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + z), $MachinePrecision] + 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

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

   1. +-commutative N/A

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

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

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

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

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 50.9

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

5. Simplified 50.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. *-lowering-*.f64 51.6

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

8. Simplified 51.6%

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

Alternative 7: 40.2% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 4.6d+135) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.6e+135], x, N[(y * z), $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. 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 52.3

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

   5. Simplified 52.3%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 39.2

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

   8. Simplified 39.2%

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

Alternative 8: 52.4% 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 51.5

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

5. Simplified 51.5%

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

Alternative 9: 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 41.8%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))