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

Percentage Accurate: 99.8% → 99.8%

Time: 4.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 99.9%

   \[x \cdot \sin y + z \cdot \cos y \]
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. Final simplification 99.9%

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

Alternative 2: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -1 \cdot 10^{+260}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot y\right) \cdot -0.16666666666666666, y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -1e+260)
     (* x (sin y))
     (if (<= y -0.001)
       t_0
       (if (<= y 0.0029)
         (fma (fma (* (* x y) -0.16666666666666666) y x) y z)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -1e+260) {
		tmp = x * sin(y);
	} else if (y <= -0.001) {
		tmp = t_0;
	} else if (y <= 0.0029) {
		tmp = fma(fma(((x * y) * -0.16666666666666666), y, x), y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -1e+260)
		tmp = Float64(x * sin(y));
	elseif (y <= -0.001)
		tmp = t_0;
	elseif (y <= 0.0029)
		tmp = fma(fma(Float64(Float64(x * y) * -0.16666666666666666), y, x), y, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e+260], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -0.001], t$95$0, If[LessEqual[y, 0.0029], N[(N[(N[(N[(x * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000007e260

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. unpow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lower-sqrt.f64 53.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 53.3

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 53.3

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

   7. Applied rewrites 53.3%

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

   8. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 82.2

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

   10. Applied rewrites 82.2%

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

   if -1.00000000000000007e260 < y < -1e-3 or 0.0029 < y

   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 60.6

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

   5. Applied rewrites 60.6%

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

   if -1e-3 < y < 0.0029

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+260}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -0.001:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot y\right) \cdot -0.16666666666666666, y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.3e+18) t_0 (if (<= z 6.2e+31) (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 <= -2.3e+18) {
		tmp = t_0;
	} else if (z <= 6.2e+31) {
		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 <= -2.3e+18)
		tmp = t_0;
	elseif (z <= 6.2e+31)
		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, -2.3e+18], t$95$0, If[LessEqual[z, 6.2e+31], 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 < -2.3e18 or 6.2000000000000004e31 < z

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 93.0

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

   5. Applied rewrites 93.0%

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

   if -2.3e18 < z < 6.2000000000000004e31

   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 92.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot y\right) \cdot -0.16666666666666666, y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -0.001)
     t_0
     (if (<= y 0.0029)
       (fma (fma (* (* x y) -0.16666666666666666) y x) y z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -0.001) {
		tmp = t_0;
	} else if (y <= 0.0029) {
		tmp = fma(fma(((x * y) * -0.16666666666666666), y, x), y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -0.001)
		tmp = t_0;
	elseif (y <= 0.0029)
		tmp = fma(fma(Float64(Float64(x * y) * -0.16666666666666666), y, x), y, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.001], t$95$0, If[LessEqual[y, 0.0029], N[(N[(N[(N[(x * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-3 or 0.0029 < y

   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 57.8

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

   5. Applied rewrites 57.8%

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

   if -1e-3 < y < 0.0029

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.001:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot y\right) \cdot -0.16666666666666666, y, x\right), y, z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.48 \cdot 10^{+228}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+203}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.48e+228) (* x y) (if (<= x 8.6e+203) (* 1.0 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.48e+228) {
		tmp = x * y;
	} else if (x <= 8.6e+203) {
		tmp = 1.0 * z;
	} else {
		tmp = x * 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 (x <= (-1.48d+228)) then
        tmp = x * y
    else if (x <= 8.6d+203) then
        tmp = 1.0d0 * z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.48e+228) {
		tmp = x * y;
	} else if (x <= 8.6e+203) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.48e+228:
		tmp = x * y
	elif x <= 8.6e+203:
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.48e+228)
		tmp = Float64(x * y);
	elseif (x <= 8.6e+203)
		tmp = Float64(1.0 * z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.48e+228)
		tmp = x * y;
	elseif (x <= 8.6e+203)
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.48e+228], N[(x * y), $MachinePrecision], If[LessEqual[x, 8.6e+203], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.48e228 or 8.6e203 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + x \cdot y} \]
   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 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.9%

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

   if -1.48e228 < x < 8.6e203

   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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 95.5

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.4%

       \[\leadsto 1 \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.0% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \sin y + z \cdot \cos y \]
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 58.9

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

5. Applied rewrites 58.9%

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

Alternative 7: 17.2% 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 99.9%

   \[x \cdot \sin y + z \cdot \cos y \]
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 58.9

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

5. Applied rewrites 58.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 16.8%

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

Reproduce

herbie shell --seed 2024308 
(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))))