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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-25} \lor \neg \left(x \leq 2.05 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.5e-25) (not (<= x 2.05e-83)))
   (fma 1.0 z (* x (sin y)))
   (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.5e-25) || !(x <= 2.05e-83)) {
		tmp = fma(1.0, z, (x * sin(y)));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.5e-25) || !(x <= 2.05e-83))
		tmp = fma(1.0, z, Float64(x * sin(y)));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.5e-25], N[Not[LessEqual[x, 2.05e-83]], $MachinePrecision]], N[(1.0 * z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5000000000000002e-25 or 2.05e-83 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 90.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 90.0

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

   7. Applied rewrites 90.0%

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

   if -3.5000000000000002e-25 < x < 2.05e-83

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-25} \lor \neg \left(x \leq 2.05 \cdot 10^{-83}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -340:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 45000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, x\right), y, z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+151}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -340.0)
     t_0
     (if (<= y 45000.0)
       (fma (fma (* -0.5 y) z x) y z)
       (if (<= y 3.1e+151) (* (cos y) z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -340.0) {
		tmp = t_0;
	} else if (y <= 45000.0) {
		tmp = fma(fma((-0.5 * y), z, x), y, z);
	} else if (y <= 3.1e+151) {
		tmp = cos(y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * x)
	tmp = 0.0
	if (y <= -340.0)
		tmp = t_0;
	elseif (y <= 45000.0)
		tmp = fma(fma(Float64(-0.5 * y), z, x), y, z);
	elseif (y <= 3.1e+151)
		tmp = Float64(cos(y) * 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, -340.0], t$95$0, If[LessEqual[y, 45000.0], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + x), $MachinePrecision] * y + z), $MachinePrecision], If[LessEqual[y, 3.1e+151], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -340 or 3.1000000000000002e151 < y

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 85.8

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.9%

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

   if -340 < y < 45000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 45000 < y < 3.1000000000000002e151

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 70.4

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

   5. Applied rewrites 70.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -340:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq 45000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, x\right), y, z\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+151}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0106 \lor \neg \left(y \leq 45000\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, x\right), y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0106) (not (<= y 45000.0)))
   (* (cos y) z)
   (fma (fma (* -0.5 y) z x) y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0106) || !(y <= 45000.0)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(fma((-0.5 * y), z, x), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0106) || !(y <= 45000.0))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(fma(Float64(-0.5 * y), z, x), y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0106], N[Not[LessEqual[y, 45000.0]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + x), $MachinePrecision] * y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0106 or 45000 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if -0.0106 < y < 45000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0106 \lor \neg \left(y \leq 45000\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, x\right), y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-187} \lor \neg \left(z \leq 2.8 \cdot 10^{-125}\right):\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5e-187) (not (<= z 2.8e-125))) (* 1.0 z) (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e-187) || !(z <= 2.8e-125)) {
		tmp = 1.0 * z;
	} else {
		tmp = x * 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 <= (-6.5d-187)) .or. (.not. (z <= 2.8d-125))) then
        tmp = 1.0d0 * z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e-187) || !(z <= 2.8e-125)) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5e-187) or not (z <= 2.8e-125):
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5e-187) || !(z <= 2.8e-125))
		tmp = Float64(1.0 * z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.5e-187) || ~((z <= 2.8e-125)))
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5e-187], N[Not[LessEqual[z, 2.8e-125]], $MachinePrecision]], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.49999999999999983e-187 or 2.8e-125 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 51.7%

      \[\leadsto 1 \cdot z \]

   if -6.49999999999999983e-187 < z < 2.8e-125

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 30.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-187} \lor \neg \left(z \leq 2.8 \cdot 10^{-125}\right):\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.3% 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 54.9

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

5. Applied rewrites 54.9%

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

6. Final simplification 54.9%

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

Alternative 7: 16.7% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * y), $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 54.9

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

5. Applied rewrites 54.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 13.7%

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

9. Final simplification 13.7%

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

Reproduce

herbie shell --seed 2024307 
(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))))