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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 8

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 2.7959578280599326e-308
  2. y = 5.103077417905526e-32
  3. z = -5.987650150757388e+161

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\sin y}{x}, z, 1\right) \cdot x\\ \mathbf{if}\;z \leq -6.9 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma (/ (sin y) x) z 1.0) x)))
   (if (<= z -6.9e+16) t_0 (if (<= z 2.35e-32) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((sin(y) / x), z, 1.0) * x;
	double tmp;
	if (z <= -6.9e+16) {
		tmp = t_0;
	} else if (z <= 2.35e-32) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(Float64(sin(y) / x), z, 1.0) * x)
	tmp = 0.0
	if (z <= -6.9e+16)
		tmp = t_0;
	elseif (z <= 2.35e-32)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] * z + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -6.9e+16], t$95$0, If[LessEqual[z, 2.35e-32], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9e16 or 2.3500000000000001e-32 < z

   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. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 54.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 54.9

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 54.9

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 83.1

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

   10. Applied rewrites 83.1%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 74.0%

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

   if -6.9e16 < z < 2.3500000000000001e-32

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 85.0

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

   5. Applied rewrites 85.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin y}{x}, z, 1\right) \cdot x\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin y}{x}, z, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;x \leq -1.42 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= x -1.42e-83) t_0 (if (<= x 1.3e-39) (* z (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (x <= -1.42e-83) {
		tmp = t_0;
	} else if (x <= 1.3e-39) {
		tmp = z * sin(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 * cos(y)
    if (x <= (-1.42d-83)) then
        tmp = t_0
    else if (x <= 1.3d-39) then
        tmp = z * sin(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 * Math.cos(y);
	double tmp;
	if (x <= -1.42e-83) {
		tmp = t_0;
	} else if (x <= 1.3e-39) {
		tmp = z * Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.cos(y)
	tmp = 0
	if x <= -1.42e-83:
		tmp = t_0
	elif x <= 1.3e-39:
		tmp = z * math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (x <= -1.42e-83)
		tmp = t_0;
	elseif (x <= 1.3e-39)
		tmp = Float64(z * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * cos(y);
	tmp = 0.0;
	if (x <= -1.42e-83)
		tmp = t_0;
	elseif (x <= 1.3e-39)
		tmp = z * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.42e-83], t$95$0, If[LessEqual[x, 1.3e-39], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999e-83 or 1.3e-39 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.4199999999999999e-83 < x < 1.3e-39

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.118:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.035)
     t_0
     (if (<= y 0.118)
       (fma (fma (fma -0.16666666666666666 (* z y) (* -0.5 x)) y z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.035) {
		tmp = t_0;
	} else if (y <= 0.118) {
		tmp = fma(fma(fma(-0.16666666666666666, (z * y), (-0.5 * x)), y, z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.035)
		tmp = t_0;
	elseif (y <= 0.118)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(z * y), Float64(-0.5 * x)), y, z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.035], t$95$0, If[LessEqual[y, 0.118], N[(N[(N[(-0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.035000000000000003 or 0.11799999999999999 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 50.7

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

   5. Applied rewrites 50.7%

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

   if -0.035000000000000003 < y < 0.11799999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.035:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.118:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.56 \cdot 10^{+103}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.56e+103) (* 1.0 x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.56e+103) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.56d+103) then
        tmp = 1.0d0 * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.56e+103) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.56e+103:
		tmp = 1.0 * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.56e+103)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.56e+103)
		tmp = 1.0 * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.56e+103], N[(1.0 * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.5599999999999999e103

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 34.7%

      \[\leadsto 1 \cdot x \]

   if 1.5599999999999999e103 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 43.0%

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

Alternative 6: 53.0% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 46.5

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

5. Applied rewrites 46.5%

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

Alternative 7: 17.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 46.5

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

5. Applied rewrites 46.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 17.5%

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

Alternative 8: 2.9% accurate, 214.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

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. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 57.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 57.2

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 57.2

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

4. Applied rewrites 57.2%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 20.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 2.9%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))