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

Percentage Accurate: 99.8% → 99.8%

Time: 14.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto x \cdot \cos y - \sin y \cdot z \]
4. Add Preprocessing

Alternative 4: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -1.7e+17)
     t_0
     (if (<= y 2900.0)
       (fma
        (fma
         (* y y)
         (fma
          (* y y)
          (fma (* y y) -0.001388888888888889 0.041666666666666664)
          -0.5)
         1.0)
        x
        (* y (* z (fma 0.16666666666666666 (* y y) -1.0))))
       (if (<= y 1.95e+42) t_0 (- (* (sin y) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -1.7e+17) {
		tmp = t_0;
	} else if (y <= 2900.0) {
		tmp = fma(fma((y * y), fma((y * y), fma((y * y), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), x, (y * (z * fma(0.16666666666666666, (y * y), -1.0))));
	} else if (y <= 1.95e+42) {
		tmp = t_0;
	} else {
		tmp = -(sin(y) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.7e+17)
		tmp = t_0;
	elseif (y <= 2900.0)
		tmp = fma(fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), x, Float64(y * Float64(z * fma(0.16666666666666666, Float64(y * y), -1.0))));
	elseif (y <= 1.95e+42)
		tmp = t_0;
	else
		tmp = Float64(-Float64(sin(y) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e+17], t$95$0, If[LessEqual[y, 2900.0], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(y * N[(z * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e+42], t$95$0, (-N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e17 or 2900 < y < 1.94999999999999985e42

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 65.5

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

   5. Applied rewrites 65.5%

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

   if -1.7e17 < y < 2900

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{0 - z \cdot \sin y}\right) \]

      3. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\frac{0 \cdot 0 - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{0 + z \cdot \sin y}}\right) \]

   6. Applied rewrites 34.6%

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\frac{\mathsf{fma}\left(\cos \left(y + y\right), -0.5, 0.5\right) \cdot \left(-z \cdot z\right)}{0 + \sin y \cdot z}}\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{y \cdot \left(-1 \cdot z + -1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{y \cdot \left(-1 \cdot z + -1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \color{blue}{\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right) + -1 \cdot z\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(-1 \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right) \cdot {y}^{2}\right)} + -1 \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right) \cdot {y}^{2}} + -1 \cdot z\right)\right) \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(\left(-1 \cdot \color{blue}{\left(z \cdot \left(\frac{-1}{3} - \frac{-1}{6}\right)\right)}\right) \cdot {y}^{2} + -1 \cdot z\right)\right) \]

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 97.7

          \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, -1\right)\right)\right) \]

   9. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)}\right) \]

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 97.8

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \]

   12. Applied rewrites 97.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \]

   if 1.94999999999999985e42 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z \cdot \sin y\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z \cdot \sin y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{z \cdot \sin y}\right) \]

      4. lower-sin.f64 71.7

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

   5. Applied rewrites 71.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 2900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 1 - \sin y \cdot z\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-178}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x 1.0) (* (sin y) z))))
   (if (<= z -9.2e-59) t_0 (if (<= z 2.7e-178) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 1.0) - (sin(y) * z);
	double tmp;
	if (z <= -9.2e-59) {
		tmp = t_0;
	} else if (z <= 2.7e-178) {
		tmp = x * cos(y);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (x * 1.0d0) - (sin(y) * z)
    if (z <= (-9.2d-59)) then
        tmp = t_0
    else if (z <= 2.7d-178) then
        tmp = x * cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 1.0) - (Math.sin(y) * z);
	double tmp;
	if (z <= -9.2e-59) {
		tmp = t_0;
	} else if (z <= 2.7e-178) {
		tmp = x * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 1.0) - (math.sin(y) * z)
	tmp = 0
	if z <= -9.2e-59:
		tmp = t_0
	elif z <= 2.7e-178:
		tmp = x * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 1.0) - Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -9.2e-59)
		tmp = t_0;
	elseif (z <= 2.7e-178)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 1.0) - (sin(y) * z);
	tmp = 0.0;
	if (z <= -9.2e-59)
		tmp = t_0;
	elseif (z <= 2.7e-178)
		tmp = x * cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 1.0), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e-59], t$95$0, If[LessEqual[z, 2.7e-178], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.19999999999999918e-59 or 2.70000000000000009e-178 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]
   4. Step-by-step derivation

   5. Applied rewrites 85.0%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \sin y \]

   if -9.19999999999999918e-59 < z < 2.70000000000000009e-178

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 92.2

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

   5. Applied rewrites 92.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-59}:\\ \;\;\;\;x \cdot 1 - \sin y \cdot z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-178}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -1.7e+17)
     t_0
     (if (<= y 2900.0)
       (fma
        (fma
         (* y y)
         (fma
          (* y y)
          (fma (* y y) -0.001388888888888889 0.041666666666666664)
          -0.5)
         1.0)
        x
        (* y (* z (fma 0.16666666666666666 (* y y) -1.0))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -1.7e+17) {
		tmp = t_0;
	} else if (y <= 2900.0) {
		tmp = fma(fma((y * y), fma((y * y), fma((y * y), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), x, (y * (z * fma(0.16666666666666666, (y * y), -1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -1.7e+17)
		tmp = t_0;
	elseif (y <= 2900.0)
		tmp = fma(fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), x, Float64(y * Float64(z * fma(0.16666666666666666, Float64(y * y), -1.0))));
	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, -1.7e+17], t$95$0, If[LessEqual[y, 2900.0], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(y * N[(z * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e17 or 2900 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 50.8

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

   5. Applied rewrites 50.8%

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

   if -1.7e17 < y < 2900

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{0 - z \cdot \sin y}\right) \]

      3. flip-- N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\frac{0 \cdot 0 - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{0 + z \cdot \sin y}}\right) \]

   6. Applied rewrites 34.6%

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{\frac{\mathsf{fma}\left(\cos \left(y + y\right), -0.5, 0.5\right) \cdot \left(-z \cdot z\right)}{0 + \sin y \cdot z}}\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{y \cdot \left(-1 \cdot z + -1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{y \cdot \left(-1 \cdot z + -1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \color{blue}{\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right) + -1 \cdot z\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(-1 \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right) \cdot {y}^{2}\right)} + -1 \cdot z\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(\color{blue}{\left(-1 \cdot \left(\frac{-1}{3} \cdot z - \frac{-1}{6} \cdot z\right)\right) \cdot {y}^{2}} + -1 \cdot z\right)\right) \]

      5. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(\left(-1 \cdot \color{blue}{\left(z \cdot \left(\frac{-1}{3} - \frac{-1}{6}\right)\right)}\right) \cdot {y}^{2} + -1 \cdot z\right)\right) \]

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 97.7

          \[\leadsto \mathsf{fma}\left(\cos y, x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, -1\right)\right)\right) \]

   9. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(\cos y, x, \color{blue}{y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)}\right) \]

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 97.8

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \]

   12. Applied rewrites 97.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, x, y \cdot \left(z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 39.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+114}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 6.5e+114) (* x 1.0) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.5e+114) {
		tmp = x * 1.0;
	} else {
		tmp = y * -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 (z <= 6.5d+114) then
        tmp = x * 1.0d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.5e+114) {
		tmp = x * 1.0;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 6.5e+114:
		tmp = x * 1.0
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 6.5e+114)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 6.5e+114)
		tmp = x * 1.0;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 6.5e+114], N[(x * 1.0), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.5000000000000001e114

   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. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 40.3%

       \[\leadsto x \cdot 1 \]

   if 6.5000000000000001e114 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.2%

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

Alternative 8: 51.9% accurate, 23.8× 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(Float64(-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

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

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.1

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

5. Applied rewrites 52.1%

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

7. Applied rewrites 52.1%

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

Alternative 9: 51.9% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 52.1

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

5. Applied rewrites 52.1%

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

6. Final simplification 52.1%

   \[\leadsto x - y \cdot z \]
7. Add Preprocessing

Alternative 10: 38.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 1.0;
}

Python

def code(x, y, z):
	return x * 1.0

Julia

function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x * 1.0), $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. 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. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 93.2

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

7. Applied rewrites 93.2%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 36.4%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))