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

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -0.75)
     t_0
     (if (<= y 0.48)
       (+
        (*
         x
         (fma
          (fma
           y
           (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
           -0.16666666666666666)
          (* y (* y y))
          y))
        (fma
         (* z (* y y))
         (fma
          (* y y)
          (fma y (* y -0.001388888888888889) 0.041666666666666664)
          -0.5)
         z))
       (if (<= y 3.1e+63) t_0 (* z (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -0.75) {
		tmp = t_0;
	} else if (y <= 0.48) {
		tmp = (x * fma(fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (y * (y * y)), y)) + fma((z * (y * y)), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), z);
	} else if (y <= 3.1e+63) {
		tmp = t_0;
	} else {
		tmp = z * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -0.75)
		tmp = t_0;
	elseif (y <= 0.48)
		tmp = Float64(Float64(x * fma(fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(y * Float64(y * y)), y)) + fma(Float64(z * Float64(y * y)), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), z));
	elseif (y <= 3.1e+63)
		tmp = t_0;
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.75], t$95$0, If[LessEqual[y, 0.48], N[(N[(x * N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+63], t$95$0, N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.75 or 0.47999999999999998 < y < 3.1000000000000001e63

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if -0.75 < y < 0.47999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \sin y + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \sin y + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) + z\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto x \cdot \sin y + \color{blue}{\mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y + 1 \cdot y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      5. *-lft-identity N/A

         \[\leadsto x \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left({y}^{2} \cdot y\right) + \color{blue}{y}\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2} \cdot y, y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

   8. Applied rewrites 99.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right) \]

   if 3.1000000000000001e63 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 72.8

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

   5. Applied rewrites 72.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+63}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.6e+98) t_0 (if (<= z 5.6e-30) (fma (sin y) x (* z 1.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.6e+98) {
		tmp = t_0;
	} else if (z <= 5.6e-30) {
		tmp = fma(sin(y), x, (z * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.6e+98)
		tmp = t_0;
	elseif (z <= 5.6e-30)
		tmp = fma(sin(y), x, Float64(z * 1.0));
	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, -2.6e+98], t$95$0, If[LessEqual[z, 5.6e-30], N[(N[Sin[y], $MachinePrecision] * x + N[(z * 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e98 or 5.59999999999999977e-30 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -2.6e98 < z < 5.59999999999999977e-30

   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 89.8%

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

Alternative 5: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -0.75)
     t_0
     (if (<= y 0.48)
       (+
        (*
         x
         (fma
          (fma
           y
           (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
           -0.16666666666666666)
          (* y (* y y))
          y))
        (fma
         (* z (* y y))
         (fma
          (* y y)
          (fma y (* y -0.001388888888888889) 0.041666666666666664)
          -0.5)
         z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -0.75) {
		tmp = t_0;
	} else if (y <= 0.48) {
		tmp = (x * fma(fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (y * (y * y)), y)) + fma((z * (y * y)), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -0.75)
		tmp = t_0;
	elseif (y <= 0.48)
		tmp = Float64(Float64(x * fma(fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(y * Float64(y * y)), y)) + fma(Float64(z * Float64(y * y)), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.75], t$95$0, If[LessEqual[y, 0.48], N[(N[(x * N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.75 or 0.47999999999999998 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if -0.75 < y < 0.47999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \sin y + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \sin y + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) + z\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto x \cdot \sin y + \color{blue}{\mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y + 1 \cdot y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      5. *-lft-identity N/A

         \[\leadsto x \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left({y}^{2} \cdot y\right) + \color{blue}{y}\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2} \cdot y, y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), z\right) \]

   8. Applied rewrites 99.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)} + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{-110}:\\ \;\;\;\;z \cdot 1\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-143}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.8e-110) (* z 1.0) (if (<= z 7.1e-143) (* y x) (* z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.8e-110) {
		tmp = z * 1.0;
	} else if (z <= 7.1e-143) {
		tmp = y * x;
	} else {
		tmp = z * 1.0;
	}
	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) :: tmp
    if (z <= (-9.8d-110)) then
        tmp = z * 1.0d0
    else if (z <= 7.1d-143) then
        tmp = y * x
    else
        tmp = z * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.8e-110) {
		tmp = z * 1.0;
	} else if (z <= 7.1e-143) {
		tmp = y * x;
	} else {
		tmp = z * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.8e-110:
		tmp = z * 1.0
	elif z <= 7.1e-143:
		tmp = y * x
	else:
		tmp = z * 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.8e-110)
		tmp = Float64(z * 1.0);
	elseif (z <= 7.1e-143)
		tmp = Float64(y * x);
	else
		tmp = Float64(z * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.8e-110)
		tmp = z * 1.0;
	elseif (z <= 7.1e-143)
		tmp = y * x;
	else
		tmp = z * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.8e-110], N[(z * 1.0), $MachinePrecision], If[LessEqual[z, 7.1e-143], N[(y * x), $MachinePrecision], N[(z * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.7999999999999995e-110 or 7.10000000000000033e-143 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.0%

      \[\leadsto z \cdot 1 \]

   if -9.7999999999999995e-110 < z < 7.10000000000000033e-143

   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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 36.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{-110}:\\ \;\;\;\;z \cdot 1\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{-143}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.0% 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 51.5

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

5. Applied rewrites 51.5%

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

Alternative 8: 17.2% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * x), $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 51.5

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

5. Applied rewrites 51.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 17.0%

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

9. Final simplification 17.0%

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

Reproduce

herbie shell --seed 2024220 
(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))))