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

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

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

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

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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * x + N[(z * N[Sin[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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+129}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin y}{x}, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.6e+129)
   (* z (sin y))
   (if (<= z 3.4e-62) (* x (cos y)) (* (fma (/ (sin y) x) z 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.6e+129) {
		tmp = z * sin(y);
	} else if (z <= 3.4e-62) {
		tmp = x * cos(y);
	} else {
		tmp = fma((sin(y) / x), z, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.6e+129)
		tmp = Float64(z * sin(y));
	elseif (z <= 3.4e-62)
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(fma(Float64(sin(y) / x), z, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.6e+129], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-62], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] * z + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.59999999999999981e129

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -4.59999999999999981e129 < z < 3.39999999999999988e-62

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

   if 3.39999999999999988e-62 < z

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. unpow1 N/A

         \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\sin y, z, \cos y \cdot x\right)\right)}^{1}} \]

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 41.8

          \[\leadsto {\left({\left(\mathsf{fma}\left(z, \sin y, x \cdot \cos y\right)\right)}^{-0.5}\right)}^{\color{blue}{-2}} \]

   6. Applied rewrites 41.8%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \sin y, x \cdot \cos y\right)\right)}^{-0.5}\right)}^{-2}} \]

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 93.9

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 75.9%

       \[\leadsto \mathsf{fma}\left(\frac{\sin y}{x}, z, 1\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+129}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin y}{x}, z, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.45:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 90000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.45)
   (* z (sin y))
   (if (<= y 90000.0) (fma (fma (* x y) -0.5 z) y x) (* x (cos y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.45) {
		tmp = z * sin(y);
	} else if (y <= 90000.0) {
		tmp = fma(fma((x * y), -0.5, z), y, x);
	} else {
		tmp = x * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.45)
		tmp = Float64(z * sin(y));
	elseif (y <= 90000.0)
		tmp = fma(fma(Float64(x * y), -0.5, z), y, x);
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.45], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 90000.0], N[(N[(N[(x * y), $MachinePrecision] * -0.5 + z), $MachinePrecision] * y + x), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.450000000000000011

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if -0.450000000000000011 < y < 9e4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 9e4 < 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 61.1

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

   5. Applied rewrites 61.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.45:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 90000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.0155:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 90000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.0155)
     t_0
     (if (<= y 90000.0) (fma (fma (* x y) -0.5 z) y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.0155) {
		tmp = t_0;
	} else if (y <= 90000.0) {
		tmp = fma(fma((x * y), -0.5, 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 <= -0.0155)
		tmp = t_0;
	elseif (y <= 90000.0)
		tmp = fma(fma(Float64(x * y), -0.5, 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, -0.0155], t$95$0, If[LessEqual[y, 90000.0], N[(N[(N[(x * y), $MachinePrecision] * -0.5 + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0155 or 9e4 < 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 49.2

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

   5. Applied rewrites 49.2%

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

   if -0.0155 < y < 9e4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0155:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 90000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% 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. lower-fma.f64 55.1

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

5. Applied rewrites 55.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. unpow1 N/A

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\sin y, z, \cos y \cdot x\right)\right)}^{1}} \]

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. sqr-pow N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 49.2

       \[\leadsto {\left({\left(\mathsf{fma}\left(z, \sin y, x \cdot \cos y\right)\right)}^{-0.5}\right)}^{\color{blue}{-2}} \]

6. Applied rewrites 49.2%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(z, \sin y, x \cdot \cos y\right)\right)}^{-0.5}\right)}^{-2}} \]

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 93.8

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

9. Applied rewrites 93.8%

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

10. Taylor expanded in y around 0

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

12. Applied rewrites 45.3%

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

Alternative 8: 17.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot \cos y + z \cdot \sin y \]
2. 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. lower-fma.f64 55.1

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

5. Applied rewrites 55.1%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 13.9%

   \[\leadsto z \cdot \color{blue}{y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))