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

Percentage Accurate: 99.8% → 99.8%

Time: 9.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(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 * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(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 * sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+230}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -0.0245:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot y, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -3.6e+230)
     (* z (cos y))
     (if (<= y -0.0245)
       t_0
       (if (<= y 2e-9)
         (fma (fma (fma -0.16666666666666666 (* x y) (* -0.5 z)) y x) y z)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -3.6e+230) {
		tmp = z * cos(y);
	} else if (y <= -0.0245) {
		tmp = t_0;
	} else if (y <= 2e-9) {
		tmp = fma(fma(fma(-0.16666666666666666, (x * y), (-0.5 * z)), y, x), y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -3.6e+230)
		tmp = Float64(z * cos(y));
	elseif (y <= -0.0245)
		tmp = t_0;
	elseif (y <= 2e-9)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(x * y), Float64(-0.5 * z)), y, x), y, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.6e+230], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -0.0245], t$95$0, If[LessEqual[y, 2e-9], N[(N[(N[(-0.16666666666666666 * N[(x * y), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.60000000000000019e230

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -3.60000000000000019e230 < y < -0.024500000000000001 or 2.00000000000000012e-9 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.8

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

   5. Applied rewrites 58.8%

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

   if -0.024500000000000001 < y < 2.00000000000000012e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+230}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -0.0245:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot y, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1 \cdot 10^{+207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1e+207) t_0 (if (<= z 5.6e+151) (fma 1.0 z (* x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1e+207) {
		tmp = t_0;
	} else if (z <= 5.6e+151) {
		tmp = fma(1.0, z, (x * sin(y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1e+207)
		tmp = t_0;
	elseif (z <= 5.6e+151)
		tmp = fma(1.0, z, Float64(x * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+207], t$95$0, If[LessEqual[z, 5.6e+151], N[(1.0 * z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e207 or 5.59999999999999975e151 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1e207 < z < 5.59999999999999975e151

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 85.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \sin y \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+207}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -2.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot y, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -2.2)
     t_0
     (if (<= y 0.031)
       (fma (fma (fma -0.16666666666666666 (* x y) (* -0.5 z)) y x) y z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -2.2) {
		tmp = t_0;
	} else if (y <= 0.031) {
		tmp = fma(fma(fma(-0.16666666666666666, (x * y), (-0.5 * z)), y, x), y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -2.2)
		tmp = t_0;
	elseif (y <= 0.031)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(x * y), Float64(-0.5 * z)), y, x), y, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2], t$95$0, If[LessEqual[y, 0.031], N[(N[(N[(-0.16666666666666666 * N[(x * y), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000002 or 0.031 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 46.1

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

   5. Applied rewrites 46.1%

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

   if -2.2000000000000002 < y < 0.031

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.031:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot y, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+180}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e+180) (* x y) (if (<= x 4.8e+166) (* 1.0 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+180) {
		tmp = x * y;
	} else if (x <= 4.8e+166) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	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 (x <= (-1.25d+180)) then
        tmp = x * y
    else if (x <= 4.8d+166) then
        tmp = 1.0d0 * z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+180) {
		tmp = x * y;
	} else if (x <= 4.8e+166) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e+180:
		tmp = x * y
	elif x <= 4.8e+166:
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e+180)
		tmp = Float64(x * y);
	elseif (x <= 4.8e+166)
		tmp = Float64(1.0 * z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.25e+180)
		tmp = x * y;
	elseif (x <= 4.8e+166)
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e+180], N[(x * y), $MachinePrecision], If[LessEqual[x, 4.8e+166], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2499999999999999e180 or 4.79999999999999984e166 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 37.8%

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

   if -1.2499999999999999e180 < x < 4.79999999999999984e166

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\frac{{\left(\cos y \cdot z\right)}^{3} + {\left(\sin y \cdot x\right)}^{3}}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot x, \left(\cos y \cdot z - \sin y \cdot x\right) \cdot \left(\cos y \cdot z\right)\right)}} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 46.9%

       \[\leadsto 1 \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+180}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+166}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 52.6

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

5. Applied rewrites 52.6%

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

Alternative 7: 16.1% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 52.6

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

5. Applied rewrites 52.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 17.1%

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

9. Final simplification 17.1%

   \[\leadsto x \cdot y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))