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

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] * x + N[(N[Cos[y], $MachinePrecision] * z), $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. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ t_1 := \sin y \cdot x\\ \mathbf{if}\;y \leq -3 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -60000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, x\right), y, z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)) (t_1 (* (sin y) x)))
   (if (<= y -3e+98)
     t_0
     (if (<= y -60000000000.0)
       t_1
       (if (<= y 2.5e-12)
         (fma (fma (* z y) -0.5 x) y z)
         (if (<= y 3.1e+121) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double t_1 = sin(y) * x;
	double tmp;
	if (y <= -3e+98) {
		tmp = t_0;
	} else if (y <= -60000000000.0) {
		tmp = t_1;
	} else if (y <= 2.5e-12) {
		tmp = fma(fma((z * y), -0.5, x), y, z);
	} else if (y <= 3.1e+121) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	t_1 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -3e+98)
		tmp = t_0;
	elseif (y <= -60000000000.0)
		tmp = t_1;
	elseif (y <= 2.5e-12)
		tmp = fma(fma(Float64(z * y), -0.5, x), y, z);
	elseif (y <= 3.1e+121)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3e+98], t$95$0, If[LessEqual[y, -60000000000.0], t$95$1, If[LessEqual[y, 2.5e-12], N[(N[(N[(z * y), $MachinePrecision] * -0.5 + x), $MachinePrecision] * y + z), $MachinePrecision], If[LessEqual[y, 3.1e+121], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000001e98 or 3.10000000000000008e121 < 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 62.0

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

   5. Applied rewrites 62.0%

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

   if -3.0000000000000001e98 < y < -6e10 or 2.49999999999999985e-12 < y < 3.10000000000000008e121

   1. Initial program 99.6%

      \[x \cdot \sin y + z \cdot \cos y \]
   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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 65.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if -6e10 < y < 2.49999999999999985e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;y \leq -60000000000:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, x\right), y, z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+121}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+80} \lor \neg \left(z \leq 7.5 \cdot 10^{+121}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e+80) (not (<= z 7.5e+121)))
   (* (cos y) z)
   (fma (sin y) x (* 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+80) || !(z <= 7.5e+121)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(sin(y), x, (1.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e+80) || !(z <= 7.5e+121))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(sin(y), x, Float64(1.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e+80], N[Not[LessEqual[z, 7.5e+121]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * x + N[(1.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000004e80 or 7.49999999999999965e121 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-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 90.1

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

   5. Applied rewrites 90.1%

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

   if -2.30000000000000004e80 < z < 7.49999999999999965e121

   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 86.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+80} \lor \neg \left(z \leq 7.5 \cdot 10^{+121}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.9 \lor \neg \left(y \leq 520\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot x, y \cdot y, x\right) \cdot y + \mathsf{fma}\left(-0.5 \cdot y, y, 1\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.9) (not (<= y 520.0)))
   (* (cos y) z)
   (+
    (*
     (fma
      (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) x)
      (* y y)
      x)
     y)
    (* (fma (* -0.5 y) y 1.0) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.9) || !(y <= 520.0)) {
		tmp = cos(y) * z;
	} else {
		tmp = (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * x), (y * y), x) * y) + (fma((-0.5 * y), y, 1.0) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.9) || !(y <= 520.0))
		tmp = Float64(cos(y) * z);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * x), Float64(y * y), x) * y) + Float64(fma(Float64(-0.5 * y), y, 1.0) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.9], N[Not[LessEqual[y, 520.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(-0.5 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.900000000000000022 or 520 < 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 52.9

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

   5. Applied rewrites 52.9%

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

   if -0.900000000000000022 < y < 520

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.008333333333333333, -0.16666666666666666 \cdot x\right), \color{blue}{y \cdot y}, x\right) \cdot y + \mathsf{fma}\left(-0.5 \cdot y, y, 1\right) \cdot z \]

   8. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.008333333333333333, -0.16666666666666666 \cdot x\right), y \cdot y, x\right) \cdot y} + \mathsf{fma}\left(-0.5 \cdot y, y, 1\right) \cdot z \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.9 \lor \neg \left(y \leq 520\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot x, y \cdot y, x\right) \cdot y + \mathsf{fma}\left(-0.5 \cdot y, y, 1\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-66} \lor \neg \left(z \leq 4.4 \cdot 10^{-201}\right):\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e-66) (not (<= z 4.4e-201))) (* 1.0 z) (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-66) || !(z <= 4.4e-201)) {
		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 ((z <= (-1.25d-66)) .or. (.not. (z <= 4.4d-201))) 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 ((z <= -1.25e-66) || !(z <= 4.4e-201)) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e-66) or not (z <= 4.4e-201):
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e-66) || !(z <= 4.4e-201))
		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 ((z <= -1.25e-66) || ~((z <= 4.4e-201)))
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e-66], N[Not[LessEqual[z, 4.4e-201]], $MachinePrecision]], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2499999999999999e-66 or 4.4e-201 < 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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 46.4%

       \[\leadsto 1 \cdot z \]

   if -1.2499999999999999e-66 < z < 4.4e-201

   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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-66} \lor \neg \left(z \leq 4.4 \cdot 10^{-201}\right):\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% 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 53.3

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

5. Applied rewrites 53.3%

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

6. Final simplification 53.3%

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

Alternative 7: 17.3% 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. 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 53.3

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

5. Applied rewrites 53.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 19.2%

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

9. Final simplification 19.2%

   \[\leadsto x \cdot y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))