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

Percentage Accurate: 99.8% → 99.8%

Time: 4.7s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y + 1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9e-62)
   (fma (sin y) z (* 1.0 x))
   (if (<= z 1.2e-20) (* (cos y) x) (+ (* z (sin y)) (* 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e-62) {
		tmp = fma(sin(y), z, (1.0 * x));
	} else if (z <= 1.2e-20) {
		tmp = cos(y) * x;
	} else {
		tmp = (z * sin(y)) + (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9e-62)
		tmp = fma(sin(y), z, Float64(1.0 * x));
	elseif (z <= 1.2e-20)
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(Float64(z * sin(y)) + Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9e-62], N[(N[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e-20], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.89999999999999986e-62

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 94.1%

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

   if -2.89999999999999986e-62 < z < 1.19999999999999996e-20

   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. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \cos y \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 91.4%

       \[\leadsto \cos y \cdot x \]

   if 1.19999999999999996e-20 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.6%

      \[\leadsto x \cdot \color{blue}{1} + z \cdot \sin y \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y + 1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+259}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.012:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -1.05e+259)
     (* (cos y) x)
     (if (<= y -0.001)
       t_0
       (if (<= y 0.012) (fma (fma (* x y) -0.5 z) y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -1.05e+259) {
		tmp = cos(y) * x;
	} else if (y <= -0.001) {
		tmp = t_0;
	} else if (y <= 0.012) {
		tmp = fma(fma((x * y), -0.5, z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -1.05e+259)
		tmp = Float64(cos(y) * x);
	elseif (y <= -0.001)
		tmp = t_0;
	elseif (y <= 0.012)
		tmp = fma(fma(Float64(x * y), -0.5, z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+259], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, -0.001], t$95$0, If[LessEqual[y, 0.012], N[(N[(N[(x * y), $MachinePrecision] * -0.5 + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000003e259

   1. Initial program 99.3%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.6

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 86.2

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

   9. Applied rewrites 86.2%

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

   10. Taylor expanded in x around inf

       \[\leadsto \cos y \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 82.1%

       \[\leadsto \cos y \cdot x \]

   if -1.05000000000000003e259 < y < -1e-3 or 0.012 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if -1e-3 < y < 0.012

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+259}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq -0.001:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.012:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) z (* 1.0 x))))
   (if (<= z -2.9e-62) t_0 (if (<= z 1.2e-20) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), z, (1.0 * x));
	double tmp;
	if (z <= -2.9e-62) {
		tmp = t_0;
	} else if (z <= 1.2e-20) {
		tmp = cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), z, Float64(1.0 * x))
	tmp = 0.0
	if (z <= -2.9e-62)
		tmp = t_0;
	elseif (z <= 1.2e-20)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.9e-62], t$95$0, If[LessEqual[z, 1.2e-20], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.89999999999999986e-62 or 1.19999999999999996e-20 < z

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.9%

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

   if -2.89999999999999986e-62 < z < 1.19999999999999996e-20

   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. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \cos y \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 91.4%

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

Alternative 5: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.012:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -0.001)
     t_0
     (if (<= y 0.012) (fma (fma (* x y) -0.5 z) y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -0.001) {
		tmp = t_0;
	} else if (y <= 0.012) {
		tmp = fma(fma((x * y), -0.5, z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -0.001)
		tmp = t_0;
	elseif (y <= 0.012)
		tmp = fma(fma(Float64(x * y), -0.5, z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.001], t$95$0, If[LessEqual[y, 0.012], N[(N[(N[(x * y), $MachinePrecision] * -0.5 + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-3 or 0.012 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if -1e-3 < y < 0.012

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.001:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.012:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot y, -0.5, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+151}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e+18) (* z y) (if (<= z 2.4e+151) (* 1.0 x) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+18) {
		tmp = z * y;
	} else if (z <= 2.4e+151) {
		tmp = 1.0 * x;
	} else {
		tmp = z * 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 <= (-2.2d+18)) then
        tmp = z * y
    else if (z <= 2.4d+151) then
        tmp = 1.0d0 * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+18) {
		tmp = z * y;
	} else if (z <= 2.4e+151) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2e+18:
		tmp = z * y
	elif z <= 2.4e+151:
		tmp = 1.0 * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e+18)
		tmp = Float64(z * y);
	elseif (z <= 2.4e+151)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2e+18)
		tmp = z * y;
	elseif (z <= 2.4e+151)
		tmp = 1.0 * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e+18], N[(z * y), $MachinePrecision], If[LessEqual[z, 2.4e+151], N[(1.0 * x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e18 or 2.4000000000000001e151 < 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 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.9%

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

   if -2.2e18 < z < 2.4000000000000001e151

   1. Initial program 99.9%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 99.7

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 97.4

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 54.4%

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

Alternative 7: 51.8% 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 58.8

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

5. Applied rewrites 58.8%

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

Alternative 8: 16.9% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z * y), $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 58.8

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

5. Applied rewrites 58.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 22.0%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))