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

Percentage Accurate: 99.8% → 99.8%

Time: 9.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 2: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, \frac{\sin y}{x}, 1\right) \cdot x\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-114}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+211}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma z (/ (sin y) x) 1.0) x)))
   (if (<= z -7.5e-78)
     t_0
     (if (<= z 2.5e-114)
       (* x (cos y))
       (if (<= z 5.4e+211) t_0 (* z (sin y)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (sin(y) / x), 1.0) * x;
	double tmp;
	if (z <= -7.5e-78) {
		tmp = t_0;
	} else if (z <= 2.5e-114) {
		tmp = x * cos(y);
	} else if (z <= 5.4e+211) {
		tmp = t_0;
	} else {
		tmp = z * sin(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(z, Float64(sin(y) / x), 1.0) * x)
	tmp = 0.0
	if (z <= -7.5e-78)
		tmp = t_0;
	elseif (z <= 2.5e-114)
		tmp = Float64(x * cos(y));
	elseif (z <= 5.4e+211)
		tmp = t_0;
	else
		tmp = Float64(z * sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -7.5e-78], t$95$0, If[LessEqual[z, 2.5e-114], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.4e+211], t$95$0, N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.50000000000000041e-78 or 2.49999999999999995e-114 < z < 5.3999999999999998e211

   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. flip-+ N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. flip3-- N/A

         \[\leadsto \frac{1}{x \cdot \cos y - z \cdot \sin y} \cdot \color{blue}{\frac{{\left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right)}^{3} - {\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right)}^{3}}{\left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right) \cdot \left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right) + \left(\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right) \cdot \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right) + \left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right) \cdot \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \frac{1}{x \cdot \cos y - z \cdot \sin y} \cdot \color{blue}{\frac{1}{\frac{\left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right) \cdot \left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right) + \left(\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right) \cdot \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right) + \left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right) \cdot \left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right)\right)}{{\left(\left(x \cdot \cos y\right) \cdot \left(x \cdot \cos y\right)\right)}^{3} - {\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right)}^{3}}}} \]

   4. Applied rewrites 54.1%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 93.2

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 77.3%

       \[\leadsto \mathsf{fma}\left(z, \frac{\sin y}{x}, 1\right) \cdot x \]

   if -7.50000000000000041e-78 < z < 2.49999999999999995e-114

   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 95.3

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

   5. Applied rewrites 95.3%

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

   if 5.3999999999999998e211 < z

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\sin y}{x}, 1\right) \cdot x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-114}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\sin y}{x}, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -7 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* x (cos y))))
   (if (<= y -7e+122)
     t_0
     (if (<= y -0.7)
       t_1
       (if (<= y 2e-17)
         (fma (fma (fma -0.16666666666666666 (* z y) (* -0.5 x)) y z) y x)
         (if (<= y 6.5e+46) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = x * cos(y);
	double tmp;
	if (y <= -7e+122) {
		tmp = t_0;
	} else if (y <= -0.7) {
		tmp = t_1;
	} else if (y <= 2e-17) {
		tmp = fma(fma(fma(-0.16666666666666666, (z * y), (-0.5 * x)), y, z), y, x);
	} else if (y <= 6.5e+46) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -7e+122)
		tmp = t_0;
	elseif (y <= -0.7)
		tmp = t_1;
	elseif (y <= 2e-17)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(z * y), Float64(-0.5 * x)), y, z), y, x);
	elseif (y <= 6.5e+46)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+122], t$95$0, If[LessEqual[y, -0.7], t$95$1, If[LessEqual[y, 2e-17], N[(N[(N[(-0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y + z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 6.5e+46], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.00000000000000028e122 or 2.00000000000000014e-17 < y < 6.50000000000000008e46

   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 67.4

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

   5. Applied rewrites 67.4%

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

   if -7.00000000000000028e122 < y < -0.69999999999999996 or 6.50000000000000008e46 < y

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -0.69999999999999996 < y < 2.00000000000000014e-17

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+122}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.7:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+46}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.7)
     t_0
     (if (<= y 5.4e+35)
       (fma (fma (fma -0.16666666666666666 (* z y) (* -0.5 x)) y z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.7) {
		tmp = t_0;
	} else if (y <= 5.4e+35) {
		tmp = fma(fma(fma(-0.16666666666666666, (z * y), (-0.5 * x)), y, 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 <= -0.7)
		tmp = t_0;
	elseif (y <= 5.4e+35)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(z * y), Float64(-0.5 * x)), y, 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, -0.7], t$95$0, If[LessEqual[y, 5.4e+35], N[(N[(N[(-0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.69999999999999996 or 5.40000000000000005e35 < y

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

   if -0.69999999999999996 < y < 5.40000000000000005e35

   1. Initial program 99.9%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+200}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 7.2e+200) (* 1.0 x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.2e+200) {
		tmp = 1.0 * x;
	} else {
		tmp = z * 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 (z <= 7.2d+200) then
        tmp = 1.0d0 * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.2e+200) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7.2e+200:
		tmp = 1.0 * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7.2e+200)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 7.2e+200)
		tmp = 1.0 * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7.2e+200], N[(1.0 * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.1999999999999995e200

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 43.3%

      \[\leadsto 1 \cdot x \]

   if 7.1999999999999995e200 < z

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]
   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 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.1%

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

Alternative 6: 52.6% 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: 17.3% 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 19.2%

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

Reproduce

herbie shell --seed 2024256 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))