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

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

Alternatives: 6

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-cos.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{+193}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -0.32)
     t_0
     (if (<= y 0.08)
       (fma
        (fma
         (* y y)
         (fma
          (* y y)
          (fma (* y y) -0.001388888888888889 0.041666666666666664)
          -0.5)
         1.0)
        x
        (* y z))
       (if (<= y 3.65e+193) t_0 (* (cos y) x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -0.32) {
		tmp = t_0;
	} else if (y <= 0.08) {
		tmp = fma(fma((y * y), fma((y * y), fma((y * y), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), x, (y * z));
	} else if (y <= 3.65e+193) {
		tmp = t_0;
	} else {
		tmp = cos(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -0.32)
		tmp = t_0;
	elseif (y <= 0.08)
		tmp = fma(fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), x, Float64(y * z));
	elseif (y <= 3.65e+193)
		tmp = t_0;
	else
		tmp = Float64(cos(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.32], t$95$0, If[LessEqual[y, 0.08], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.65e+193], t$95$0, N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.320000000000000007 or 0.0800000000000000017 < y < 3.6500000000000001e193

   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 58.3

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

   5. Simplified 58.3%

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

   if -0.320000000000000007 < y < 0.0800000000000000017

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}, x, y \cdot z\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1}, x, y \cdot z\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({y}^{2}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right)}, x, y \cdot z\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right), x, y \cdot z\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right), x, y \cdot z\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), x, y \cdot z\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right), x, y \cdot z\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)}, 1\right), x, y \cdot z\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right), x, y \cdot z\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right), x, y \cdot z\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right), x, y \cdot z\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right), x, y \cdot z\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right), x, y \cdot z\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right), x, y \cdot z\right) \]

      14. lower-*.f64 99.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot z\right) \]

   10. Simplified 99.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, x, y \cdot z\right) \]

   if 3.6500000000000001e193 < 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 68.6

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

   5. Simplified 68.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), x, y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{+193}:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) z x)))
   (if (<= z -3e-16) t_0 (if (<= z 8.5e-147) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), z, x);
	double tmp;
	if (z <= -3e-16) {
		tmp = t_0;
	} else if (z <= 8.5e-147) {
		tmp = cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), z, x)
	tmp = 0.0
	if (z <= -3e-16)
		tmp = t_0;
	elseif (z <= 8.5e-147)
		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 + x), $MachinePrecision]}, If[LessEqual[z, -3e-16], t$95$0, If[LessEqual[z, 8.5e-147], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999994e-16 or 8.5000000000000002e-147 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 89.1%

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

      1. *-rgt-identity 89.1

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

   9. Applied egg-rr 89.1%

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

   if -2.99999999999999994e-16 < z < 8.5000000000000002e-147

   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 89.5

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

   5. Simplified 89.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -170000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.045:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -170000.0)
     t_0
     (if (<= y 0.045)
       (fma y (fma y (fma x -0.5 (* (* y z) -0.16666666666666666)) z) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -170000.0) {
		tmp = t_0;
	} else if (y <= 0.045) {
		tmp = fma(y, fma(y, fma(x, -0.5, ((y * z) * -0.16666666666666666)), z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -170000.0)
		tmp = t_0;
	elseif (y <= 0.045)
		tmp = fma(y, fma(y, fma(x, -0.5, Float64(Float64(y * z) * -0.16666666666666666)), z), x);
	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, -170000.0], t$95$0, If[LessEqual[y, 0.045], N[(y * N[(y * N[(x * -0.5 + N[(N[(y * z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e5 or 0.044999999999999998 < 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 46.7

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

   5. Simplified 46.7%

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

   if -1.7e5 < y < 0.044999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -170000:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.045:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, -0.5, \left(y \cdot z\right) \cdot -0.16666666666666666\right), z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% 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 55.0

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

5. Simplified 55.0%

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

Alternative 6: 17.2% 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 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 40.0

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

5. Simplified 40.0%

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

6. Taylor expanded in y around 0

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

   1. lower-*.f64 15.9

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

8. Simplified 15.9%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))