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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 10

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 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. 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 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\cos y}}{z}}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+24}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0105:\\ \;\;\;\;t\_0 + y \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -4e+231)
     (/ 1.0 (/ (/ 1.0 (cos y)) z))
     (if (<= y -7.2e+24)
       (* (sin y) x)
       (if (<= y 0.0105)
         (+
          t_0
          (*
           y
           (fma
            (* x (* y y))
            (fma
             y
             (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
             -0.16666666666666666)
            x)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -4e+231) {
		tmp = 1.0 / ((1.0 / cos(y)) / z);
	} else if (y <= -7.2e+24) {
		tmp = sin(y) * x;
	} else if (y <= 0.0105) {
		tmp = t_0 + (y * fma((x * (y * y)), fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -4e+231)
		tmp = Float64(1.0 / Float64(Float64(1.0 / cos(y)) / z));
	elseif (y <= -7.2e+24)
		tmp = Float64(sin(y) * x);
	elseif (y <= 0.0105)
		tmp = Float64(t_0 + Float64(y * fma(Float64(x * Float64(y * y)), fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), x)));
	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, -4e+231], N[(1.0 / N[(N[(1.0 / N[Cos[y], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.2e+24], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 0.0105], N[(t$95$0 + N[(y * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.0000000000000002e231

   1. Initial program 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      11. lower-/.f64 99.2

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \sin y + z \cdot \cos y}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y + x \cdot \sin y}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y} + x \cdot \sin y}} \]

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      3. lower-cos.f64 67.2

         \[\leadsto \frac{1}{\frac{1}{z \cdot \color{blue}{\cos y}}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{z \cdot \color{blue}{\cos y}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      3. unpow1 N/A

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

      4. pow-flip N/A

         \[\leadsto \frac{1}{\color{blue}{{\left(z \cdot \cos y\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}} \]

      5. metadata-eval N/A

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

      6. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y \cdot z}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

      11. lower-/.f64 67.6

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{\cos y}}}{z}} \]

   9. Applied rewrites 67.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

   if -4.0000000000000002e231 < y < -7.19999999999999966e24

   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 68.0

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

   5. Applied rewrites 68.0%

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

   if -7.19999999999999966e24 < y < 0.0105000000000000007

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

   if 0.0105000000000000007 < 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 56.5

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

   5. Applied rewrites 56.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\cos y}}{z}}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+24}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0105:\\ \;\;\;\;z \cdot \cos y + y \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\cos y}}{z}}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 17.5:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -4e+231)
     (/ 1.0 (/ (/ 1.0 (cos y)) z))
     (if (<= y -4.8e+19)
       t_0
       (if (<= y 17.5)
         (+
          t_0
          (fma
           (* z (* y y))
           (fma
            (* y y)
            (fma y (* y -0.001388888888888889) 0.041666666666666664)
            -0.5)
           z))
         (* z (cos y)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -4e+231) {
		tmp = 1.0 / ((1.0 / cos(y)) / z);
	} else if (y <= -4.8e+19) {
		tmp = t_0;
	} else if (y <= 17.5) {
		tmp = t_0 + fma((z * (y * y)), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), z);
	} else {
		tmp = z * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -4e+231)
		tmp = Float64(1.0 / Float64(Float64(1.0 / cos(y)) / z));
	elseif (y <= -4.8e+19)
		tmp = t_0;
	elseif (y <= 17.5)
		tmp = Float64(t_0 + fma(Float64(z * Float64(y * y)), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), z));
	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, -4e+231], N[(1.0 / N[(N[(1.0 / N[Cos[y], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.8e+19], t$95$0, If[LessEqual[y, 17.5], N[(t$95$0 + 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], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.0000000000000002e231

   1. Initial program 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      11. lower-/.f64 99.2

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \sin y + z \cdot \cos y}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y + x \cdot \sin y}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y} + x \cdot \sin y}} \]

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      3. lower-cos.f64 67.2

         \[\leadsto \frac{1}{\frac{1}{z \cdot \color{blue}{\cos y}}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{z \cdot \color{blue}{\cos y}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      3. unpow1 N/A

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

      4. pow-flip N/A

         \[\leadsto \frac{1}{\color{blue}{{\left(z \cdot \cos y\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}} \]

      5. metadata-eval N/A

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

      6. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y \cdot z}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

      11. lower-/.f64 67.6

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{\cos y}}}{z}} \]

   9. Applied rewrites 67.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

   if -4.0000000000000002e231 < y < -4.8e19

   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 66.2

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

   5. Applied rewrites 66.2%

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

   if -4.8e19 < y < 17.5

   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 97.1%

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

   if 17.5 < 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 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\cos y}}{z}}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+19}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 17.5:\\ \;\;\;\;\sin y \cdot x + \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\cos y}}{z}}\\ \mathbf{elif}\;y \leq -0.016:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0102:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+231)
   (/ 1.0 (/ (/ 1.0 (cos y)) z))
   (if (<= y -0.016)
     (* (sin y) x)
     (if (<= y 0.0102)
       (fma y (fma y (fma z -0.5 (* -0.16666666666666666 (* y x))) x) z)
       (* z (cos y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+231) {
		tmp = 1.0 / ((1.0 / cos(y)) / z);
	} else if (y <= -0.016) {
		tmp = sin(y) * x;
	} else if (y <= 0.0102) {
		tmp = fma(y, fma(y, fma(z, -0.5, (-0.16666666666666666 * (y * x))), x), z);
	} else {
		tmp = z * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+231)
		tmp = Float64(1.0 / Float64(Float64(1.0 / cos(y)) / z));
	elseif (y <= -0.016)
		tmp = Float64(sin(y) * x);
	elseif (y <= 0.0102)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(-0.16666666666666666 * Float64(y * x))), x), z);
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+231], N[(1.0 / N[(N[(1.0 / N[Cos[y], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -0.016], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 0.0102], N[(y * N[(y * N[(z * -0.5 + N[(-0.16666666666666666 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.0000000000000002e231

   1. Initial program 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      11. lower-/.f64 99.2

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x \cdot \sin y + z \cdot \cos y}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \sin y + z \cdot \cos y}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y + x \cdot \sin y}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y} + x \cdot \sin y}} \]

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      3. lower-cos.f64 67.2

         \[\leadsto \frac{1}{\frac{1}{z \cdot \color{blue}{\cos y}}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{z \cdot \color{blue}{\cos y}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      3. unpow1 N/A

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

      4. pow-flip N/A

         \[\leadsto \frac{1}{\color{blue}{{\left(z \cdot \cos y\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}} \]

      5. metadata-eval N/A

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

      6. inv-pow N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{z \cdot \cos y}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{z \cdot \cos y}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y \cdot z}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

      11. lower-/.f64 67.6

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{\cos y}}}{z}} \]

   9. Applied rewrites 67.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{\cos y}}{z}}} \]

   if -4.0000000000000002e231 < y < -0.016

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 61.8

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

   5. Applied rewrites 61.8%

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

   if -0.016 < y < 0.010200000000000001

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 0.010200000000000001 < 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 56.5

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

   5. Applied rewrites 56.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\cos y}}{z}}\\ \mathbf{elif}\;y \leq -0.016:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0102:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+231}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.016:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0102:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -3.8e+231)
     t_0
     (if (<= y -0.016)
       (* (sin y) x)
       (if (<= y 0.0102)
         (fma y (fma y (fma z -0.5 (* -0.16666666666666666 (* y x))) x) z)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -3.8e+231) {
		tmp = t_0;
	} else if (y <= -0.016) {
		tmp = sin(y) * x;
	} else if (y <= 0.0102) {
		tmp = fma(y, fma(y, fma(z, -0.5, (-0.16666666666666666 * (y * x))), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -3.8e+231)
		tmp = t_0;
	elseif (y <= -0.016)
		tmp = Float64(sin(y) * x);
	elseif (y <= 0.0102)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(-0.16666666666666666 * Float64(y * x))), x), 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, -3.8e+231], t$95$0, If[LessEqual[y, -0.016], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 0.0102], N[(y * N[(y * N[(z * -0.5 + N[(-0.16666666666666666 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000001e231 or 0.010200000000000001 < 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 58.7

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

   5. Applied rewrites 58.7%

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

   if -3.8000000000000001e231 < y < -0.016

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 61.8

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

   5. Applied rewrites 61.8%

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

   if -0.016 < y < 0.010200000000000001

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+231}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -0.016:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.0102:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -0.016:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1850000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -0.016)
     t_0
     (if (<= y 1850000.0)
       (fma y (fma y (fma z -0.5 (* -0.16666666666666666 (* y x))) x) z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -0.016) {
		tmp = t_0;
	} else if (y <= 1850000.0) {
		tmp = fma(y, fma(y, fma(z, -0.5, (-0.16666666666666666 * (y * x))), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -0.016)
		tmp = t_0;
	elseif (y <= 1850000.0)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(-0.16666666666666666 * Float64(y * x))), x), 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, -0.016], t$95$0, If[LessEqual[y, 1850000.0], N[(y * N[(y * N[(z * -0.5 + N[(-0.16666666666666666 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.016 or 1.85e6 < 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 51.3

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

   5. Applied rewrites 51.3%

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

   if -0.016 < y < 1.85e6

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.016:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 1850000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.7% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+113}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -7.2e+113) (* y x) z))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e+113) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7.2d+113)) then
        tmp = y * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e+113) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.2e+113:
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.2e+113)
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.2e+113)
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e+113], N[(y * x), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999984e113

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 30.7

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

   8. Applied rewrites 30.7%

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

   if -7.19999999999999984e113 < x

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.1%

      \[\leadsto z \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 45.1

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

   10. Applied rewrites 45.1%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+113}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 99.7%

   \[x \cdot \sin y + z \cdot \cos y \]
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 51.9

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

5. Applied rewrites 51.9%

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

Alternative 10: 38.5% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

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 63.0

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

5. Applied rewrites 63.0%

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

6. Taylor expanded in y around 0

   \[\leadsto z \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 40.1%

   \[\leadsto z \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 40.1

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

10. Applied rewrites 40.1%

    \[\leadsto \color{blue}{z} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))