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

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ t_1 := z \cdot \cos y\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.0085:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, x \cdot \left(y \cdot -0.16666666666666666\right)\right), x\right), z\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)) (t_1 (* z (cos y))))
   (if (<= y -8.5e+217)
     t_0
     (if (<= y -0.0085)
       t_1
       (if (<= y 0.00055)
         (fma y (fma y (fma z -0.5 (* x (* y -0.16666666666666666))) x) z)
         (if (<= y 4.6e+88) t_1 (if (<= y 8.2e+245) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double t_1 = z * cos(y);
	double tmp;
	if (y <= -8.5e+217) {
		tmp = t_0;
	} else if (y <= -0.0085) {
		tmp = t_1;
	} else if (y <= 0.00055) {
		tmp = fma(y, fma(y, fma(z, -0.5, (x * (y * -0.16666666666666666))), x), z);
	} else if (y <= 4.6e+88) {
		tmp = t_1;
	} else if (y <= 8.2e+245) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	t_1 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -8.5e+217)
		tmp = t_0;
	elseif (y <= -0.0085)
		tmp = t_1;
	elseif (y <= 0.00055)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(x * Float64(y * -0.16666666666666666))), x), z);
	elseif (y <= 4.6e+88)
		tmp = t_1;
	elseif (y <= 8.2e+245)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e+217], t$95$0, If[LessEqual[y, -0.0085], t$95$1, If[LessEqual[y, 0.00055], N[(y * N[(y * N[(z * -0.5 + N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[y, 4.6e+88], t$95$1, If[LessEqual[y, 8.2e+245], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.50000000000000021e217 or 4.6000000000000003e88 < y < 8.2000000000000001e245

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if -8.50000000000000021e217 < y < -0.0085000000000000006 or 5.50000000000000033e-4 < y < 4.6000000000000003e88 or 8.2000000000000001e245 < 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. lower-*.f64 N/A

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

      2. lower-cos.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -0.0085000000000000006 < y < 5.50000000000000033e-4

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+217}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq -0.0085:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.00055:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, x \cdot \left(y \cdot -0.16666666666666666\right)\right), x\right), z\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+88}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+245}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, z \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1.5e+62) t_0 (if (<= z 1.5e+41) (fma (sin y) x (* z 1.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1.5e+62) {
		tmp = t_0;
	} else if (z <= 1.5e+41) {
		tmp = fma(sin(y), x, (z * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1.5e+62)
		tmp = t_0;
	elseif (z <= 1.5e+41)
		tmp = fma(sin(y), x, Float64(z * 1.0));
	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, -1.5e+62], t$95$0, If[LessEqual[z, 1.5e+41], N[(N[Sin[y], $MachinePrecision] * x + N[(z * 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e62 or 1.4999999999999999e41 < 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. lower-*.f64 N/A

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

      2. lower-cos.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -1.5e62 < z < 1.4999999999999999e41

   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

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 85.7%

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

Alternative 5: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -22000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, x \cdot \left(y \cdot -0.16666666666666666\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -22000000000.0)
     t_0
     (if (<= y 7000.0)
       (fma y (fma y (fma z -0.5 (* x (* y -0.16666666666666666))) x) z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -22000000000.0) {
		tmp = t_0;
	} else if (y <= 7000.0) {
		tmp = fma(y, fma(y, fma(z, -0.5, (x * (y * -0.16666666666666666))), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -22000000000.0)
		tmp = t_0;
	elseif (y <= 7000.0)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(x * Float64(y * -0.16666666666666666))), x), z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -22000000000.0], t$95$0, If[LessEqual[y, 7000.0], N[(y * N[(y * N[(z * -0.5 + N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e10 or 7e3 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 48.6

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

   5. Applied rewrites 48.6%

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

   if -2.2e10 < y < 7e3

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -22000000000:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 7000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, x \cdot \left(y \cdot -0.16666666666666666\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.0% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+148}:\\ \;\;\;\;z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 5.2e+148) (* z 1.0) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.2e+148) {
		tmp = z * 1.0;
	} else {
		tmp = y * x;
	}
	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 <= 5.2d+148) then
        tmp = z * 1.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.2e+148) {
		tmp = z * 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5.2e+148:
		tmp = z * 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.2e+148)
		tmp = Float64(z * 1.0);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5.2e+148)
		tmp = z * 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.2e+148], N[(z * 1.0), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e148

   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. lower-*.f64 N/A

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

      2. lower-cos.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.3%

      \[\leadsto z \cdot 1 \]

   if 5.2e148 < x

   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 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 47.5%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+148}:\\ \;\;\;\;z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% 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 53.0

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

5. Applied rewrites 53.0%

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

Alternative 8: 16.9% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * x), $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 53.0

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 17.8%

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

9. Final simplification 17.8%

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

Reproduce

herbie shell --seed 2024232 
(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))))