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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 6

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = -7.133505650084179e-7
  2. y = 1.339700078360316e+255
  3. z = 3.9839737876275757e-308

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1.3e-32) t_0 (if (<= z 4.8e+51) (fma 1.0 z (* x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1.3e-32) {
		tmp = t_0;
	} else if (z <= 4.8e+51) {
		tmp = fma(1.0, z, (x * sin(y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1.3e-32)
		tmp = t_0;
	elseif (z <= 4.8e+51)
		tmp = fma(1.0, z, Float64(x * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e-32], t$95$0, If[LessEqual[z, 4.8e+51], N[(1.0 * z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e-32 or 4.7999999999999997e51 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -1.2999999999999999e-32 < z < 4.7999999999999997e51

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 88.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-32}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) \cdot z + \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot x, y \cdot y, x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -65.0)
     t_0
     (if (<= y 0.48)
       (+
        (* (fma (fma 0.041666666666666664 (* y y) -0.5) (* y y) 1.0) z)
        (*
         (fma
          (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) x)
          (* y y)
          x)
         y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -65.0) {
		tmp = t_0;
	} else if (y <= 0.48) {
		tmp = (fma(fma(0.041666666666666664, (y * y), -0.5), (y * y), 1.0) * z) + (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * x), (y * y), x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -65.0)
		tmp = t_0;
	elseif (y <= 0.48)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(y * y), -0.5), Float64(y * y), 1.0) * z) + Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * x), Float64(y * y), x) * y));
	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, -65.0], t$95$0, If[LessEqual[y, 0.48], N[(N[(N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -65 or 0.47999999999999998 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if -65 < y < 0.47999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, x\right) \cdot y} + z \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) \cdot z + \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot x, y \cdot y, x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-129}:\\ \;\;\;\;1 \cdot z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-90}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7e-129) (* 1.0 z) (if (<= z 8e-90) (* x y) (* 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e-129) {
		tmp = 1.0 * z;
	} else if (z <= 8e-90) {
		tmp = x * y;
	} else {
		tmp = 1.0 * 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 <= (-7d-129)) then
        tmp = 1.0d0 * z
    else if (z <= 8d-90) then
        tmp = x * y
    else
        tmp = 1.0d0 * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e-129) {
		tmp = 1.0 * z;
	} else if (z <= 8e-90) {
		tmp = x * y;
	} else {
		tmp = 1.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7e-129:
		tmp = 1.0 * z
	elif z <= 8e-90:
		tmp = x * y
	else:
		tmp = 1.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7e-129)
		tmp = Float64(1.0 * z);
	elseif (z <= 8e-90)
		tmp = Float64(x * y);
	else
		tmp = Float64(1.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e-129)
		tmp = 1.0 * z;
	elseif (z <= 8e-90)
		tmp = x * y;
	else
		tmp = 1.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7e-129], N[(1.0 * z), $MachinePrecision], If[LessEqual[z, 8e-90], N[(x * y), $MachinePrecision], N[(1.0 * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9999999999999995e-129 or 7.99999999999999996e-90 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 48.7%

       \[\leadsto 1 \cdot z \]

   if -6.9999999999999995e-129 < z < 7.99999999999999996e-90

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.5, z, \frac{x \cdot x}{z}\right) - \frac{x \cdot x}{z}\right) - z, y, x\right), y, z\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.25 \cdot \left(z \cdot z\right) - 0}{0.5 \cdot z - 0} - z, y, x\right), y, z\right) \]

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 46.6%

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

Alternative 5: 53.3% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 55.1

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

5. Applied rewrites 55.1%

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

Alternative 6: 17.6% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. swap-sqr N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 51.8%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. remove-double-neg N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 38.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.5, z, \frac{x \cdot x}{z}\right) - \frac{x \cdot x}{z}\right) - z, y, x\right), y, z\right)} \]
8. Step-by-step derivation

9. Applied rewrites 41.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.25 \cdot \left(z \cdot z\right) - 0}{0.5 \cdot z - 0} - z, y, x\right), y, z\right) \]

10. Taylor expanded in x around inf

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

12. Applied rewrites 18.3%

    \[\leadsto x \cdot \color{blue}{y} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024297 
(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))))