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

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 74.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \cos y \cdot x\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.075:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y \cdot y, 1\right), x, t\_0\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+205}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (* (cos y) x)))
   (if (<= y -1.02e+27)
     t_1
     (if (<= y 0.075)
       (fma (fma -0.5 (* y y) 1.0) x t_0)
       (if (<= y 1.9e+205) t_0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = cos(y) * x;
	double tmp;
	if (y <= -1.02e+27) {
		tmp = t_1;
	} else if (y <= 0.075) {
		tmp = fma(fma(-0.5, (y * y), 1.0), x, t_0);
	} else if (y <= 1.9e+205) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -1.02e+27)
		tmp = t_1;
	elseif (y <= 0.075)
		tmp = fma(fma(-0.5, Float64(y * y), 1.0), x, t_0);
	elseif (y <= 1.9e+205)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -1.02e+27], t$95$1, If[LessEqual[y, 0.075], N[(N[(-0.5 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x + t$95$0), $MachinePrecision], If[LessEqual[y, 1.9e+205], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0199999999999999e27 or 1.9e205 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if -1.0199999999999999e27 < y < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 0.0749999999999999972 < y < 1.9e205

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 59.1

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

   5. Applied rewrites 59.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.075:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y \cdot y, 1\right), x, z \cdot \sin y\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+205}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -260:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.019:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot -0.5, z\right), x\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+205}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -260.0)
     t_0
     (if (<= y 0.019)
       (fma y (fma y (* x -0.5) z) x)
       (if (<= y 1.9e+205) (* z (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -260.0) {
		tmp = t_0;
	} else if (y <= 0.019) {
		tmp = fma(y, fma(y, (x * -0.5), z), x);
	} else if (y <= 1.9e+205) {
		tmp = z * sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -260.0)
		tmp = t_0;
	elseif (y <= 0.019)
		tmp = fma(y, fma(y, Float64(x * -0.5), z), x);
	elseif (y <= 1.9e+205)
		tmp = Float64(z * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -260.0], t$95$0, If[LessEqual[y, 0.019], N[(y * N[(y * N[(x * -0.5), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.9e+205], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -260 or 1.9e205 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if -260 < y < 0.0189999999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.0189999999999999995 < y < 1.9e205

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 59.1

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

   5. Applied rewrites 59.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.019:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot -0.5, z\right), x\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+205}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.55)
     t_0
     (if (<= y 0.75)
       (fma
        (fma -0.5 (* y y) 1.0)
        x
        (*
         y
         (fma
          (* y y)
          (fma
           (* y y)
           (* z (fma -0.0001984126984126984 (* y y) 0.008333333333333333))
           (* z -0.16666666666666666))
          z)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.55) {
		tmp = t_0;
	} else if (y <= 0.75) {
		tmp = fma(fma(-0.5, (y * y), 1.0), x, (y * fma((y * y), fma((y * y), (z * fma(-0.0001984126984126984, (y * y), 0.008333333333333333)), (z * -0.16666666666666666)), z)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.55)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = fma(fma(-0.5, Float64(y * y), 1.0), x, Float64(y * fma(Float64(y * y), fma(Float64(y * y), Float64(z * fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333)), Float64(z * -0.16666666666666666)), z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -0.55], t$95$0, If[LessEqual[y, 0.75], N[(N[(-0.5 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(z * N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.55000000000000004 or 0.75 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 54.9

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

   5. Applied rewrites 54.9%

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

   if -0.55000000000000004 < y < 0.75

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right) + \frac{-1}{6} \cdot z}, z\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z, \frac{-1}{6} \cdot z\right)}, z\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z, \frac{-1}{6} \cdot z\right), z\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z, \frac{-1}{6} \cdot z\right), z\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right) \cdot z} + \frac{1}{120} \cdot z, \frac{-1}{6} \cdot z\right), z\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{z \cdot \left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}, \frac{-1}{6} \cdot z\right), z\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{z \cdot \left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}, \frac{-1}{6} \cdot z\right), z\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right)}, \frac{-1}{6} \cdot z\right), z\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{y \cdot y}, \frac{1}{120}\right), \frac{-1}{6} \cdot z\right), z\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{y \cdot y}, \frac{1}{120}\right), \frac{-1}{6} \cdot z\right), z\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), \color{blue}{z \cdot \frac{-1}{6}}\right), z\right)\right) \]

      17. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), \color{blue}{z \cdot -0.16666666666666666}\right), z\right)\right) \]

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y \cdot y, 1\right), x, \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.55:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y \cdot y, 1\right), x, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+151}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+174}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.6e+151) (* y z) (if (<= z 1.95e+174) (* x 1.0) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.6e+151) {
		tmp = y * z;
	} else if (z <= 1.95e+174) {
		tmp = x * 1.0;
	} else {
		tmp = y * 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 <= (-9.6d+151)) then
        tmp = y * z
    else if (z <= 1.95d+174) then
        tmp = x * 1.0d0
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.6e+151) {
		tmp = y * z;
	} else if (z <= 1.95e+174) {
		tmp = x * 1.0;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.6e+151:
		tmp = y * z
	elif z <= 1.95e+174:
		tmp = x * 1.0
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.6e+151)
		tmp = Float64(y * z);
	elseif (z <= 1.95e+174)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.6e+151)
		tmp = y * z;
	elseif (z <= 1.95e+174)
		tmp = x * 1.0;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.6e+151], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.95e+174], N[(x * 1.0), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.6000000000000004e151 or 1.9499999999999999e174 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.0%

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

   if -9.6000000000000004e151 < z < 1.9499999999999999e174

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.9%

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

Alternative 6: 51.9% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 54.2

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

5. Applied rewrites 54.2%

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

Alternative 7: 16.9% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

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 * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 18.1%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))