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

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 27000000000000:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, \frac{\sin y}{z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.35e+100)
     t_0
     (if (<= z 27000000000000.0) (* z (fma x (/ (sin y) z) 1.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.35e+100) {
		tmp = t_0;
	} else if (z <= 27000000000000.0) {
		tmp = z * fma(x, (sin(y) / z), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.35e+100)
		tmp = t_0;
	elseif (z <= 27000000000000.0)
		tmp = Float64(z * fma(x, Float64(sin(y) / z), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.35e+100], t$95$0, If[LessEqual[z, 27000000000000.0], N[(z * N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.35e100 or 2.7e13 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 89.5

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

   5. Simplified 89.5%

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

   if -2.35e100 < z < 2.7e13

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 90.5

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

   7. Simplified 90.5%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 76.7%

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

Alternative 3: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -0.0046:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -0.0046)
     t_0
     (if (<= y 0.5)
       (fma y (fma y (fma z -0.5 (* -0.16666666666666666 (* y x))) x) z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -0.0046) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = fma(y, fma(y, fma(z, -0.5, (-0.16666666666666666 * (y * x))), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -0.0046)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = fma(y, fma(y, fma(z, -0.5, Float64(-0.16666666666666666 * Float64(y * x))), x), z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0046], t$95$0, If[LessEqual[y, 0.5], N[(y * N[(y * N[(z * -0.5 + N[(-0.16666666666666666 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0045999999999999999 or 0.5 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 59.9

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

   5. Simplified 59.9%

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

   if -0.0045999999999999999 < y < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0046:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, -0.16666666666666666 \cdot \left(y \cdot x\right)\right), x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.145:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot -0.16666666666666666, z \cdot -0.5\right) \cdot \left(y \cdot y\right) + \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -75000000.0)
     t_0
     (if (<= y 0.145)
       (+
        (* (fma y (* x -0.16666666666666666) (* z -0.5)) (* y y))
        (fma y x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -75000000.0) {
		tmp = t_0;
	} else if (y <= 0.145) {
		tmp = (fma(y, (x * -0.16666666666666666), (z * -0.5)) * (y * y)) + fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -75000000.0)
		tmp = t_0;
	elseif (y <= 0.145)
		tmp = Float64(Float64(fma(y, Float64(x * -0.16666666666666666), Float64(z * -0.5)) * Float64(y * y)) + fma(y, x, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -75000000.0], t$95$0, If[LessEqual[y, 0.145], N[(N[(N[(y * N[(x * -0.16666666666666666), $MachinePrecision] + N[(z * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(y * x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e7 or 0.14499999999999999 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 43.5

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

   5. Simplified 43.5%

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

   if -7.5e7 < y < 0.14499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.0

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

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.5, \left(y \cdot x\right) \cdot -0.16666666666666666\right), x\right), z\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

   7. Applied egg-rr 97.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 0.145:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot -0.16666666666666666, z \cdot -0.5\right) \cdot \left(y \cdot y\right) + \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+165}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+228}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e+165) (* y x) (if (<= x 2.55e+228) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+165) {
		tmp = y * x;
	} else if (x <= 2.55e+228) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.3d+165)) then
        tmp = y * x
    else if (x <= 2.55d+228) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+165) {
		tmp = y * x;
	} else if (x <= 2.55e+228) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e+165:
		tmp = y * x
	elif x <= 2.55e+228:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e+165)
		tmp = Float64(y * x);
	elseif (x <= 2.55e+228)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e+165)
		tmp = y * x;
	elseif (x <= 2.55e+228)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e+165], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.55e+228], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3000000000000001e165 or 2.54999999999999986e228 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 57.6

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

   5. Simplified 57.6%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 43.5

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

   8. Simplified 43.5%

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

   if -1.3000000000000001e165 < x < 2.54999999999999986e228

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 78.6

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

   5. Simplified 78.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 47.6%

      \[\leadsto z \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 47.6

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

   10. Applied egg-rr 47.6%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+165}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+228}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 54.5

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

5. Simplified 54.5%

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

Alternative 7: 38.7% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 67.1

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

5. Simplified 67.1%

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

6. Taylor expanded in y around 0

   \[\leadsto z \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 41.7%

   \[\leadsto z \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 41.7

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

10. Applied egg-rr 41.7%

    \[\leadsto \color{blue}{z} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))