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

Percentage Accurate: 99.8% → 99.8%

Time: 12.1s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + N[(x * N[Sin[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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ t_1 := x \cdot \sin y\\ t_2 := z \cdot \mathsf{fma}\left(t\_1, \frac{1}{z}, 1\right)\\ \mathbf{if}\;z \leq -1.06 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-259}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{-255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z))
        (t_1 (* x (sin y)))
        (t_2 (* z (fma t_1 (/ 1.0 z) 1.0))))
   (if (<= z -1.06e+75)
     t_0
     (if (<= z -2.2e-259)
       t_2
       (if (<= z 3.95e-255) t_1 (if (<= z 4.2e-40) t_2 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double t_1 = x * sin(y);
	double t_2 = z * fma(t_1, (1.0 / z), 1.0);
	double tmp;
	if (z <= -1.06e+75) {
		tmp = t_0;
	} else if (z <= -2.2e-259) {
		tmp = t_2;
	} else if (z <= 3.95e-255) {
		tmp = t_1;
	} else if (z <= 4.2e-40) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	t_1 = Float64(x * sin(y))
	t_2 = Float64(z * fma(t_1, Float64(1.0 / z), 1.0))
	tmp = 0.0
	if (z <= -1.06e+75)
		tmp = t_0;
	elseif (z <= -2.2e-259)
		tmp = t_2;
	elseif (z <= 3.95e-255)
		tmp = t_1;
	elseif (z <= 4.2e-40)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t$95$1 * N[(1.0 / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.06e+75], t$95$0, If[LessEqual[z, -2.2e-259], t$95$2, If[LessEqual[z, 3.95e-255], t$95$1, If[LessEqual[z, 4.2e-40], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.06e75 or 4.20000000000000036e-40 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   if -1.06e75 < z < -2.2000000000000001e-259 or 3.95000000000000005e-255 < z < 4.20000000000000036e-40

   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. 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. lower-/.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 88.3

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

   7. Applied rewrites 88.3%

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

   9. Applied rewrites 88.4%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 84.1%

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

   if -2.2000000000000001e-259 < z < 3.95000000000000005e-255

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+75}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-259}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x \cdot \sin y, \frac{1}{z}, 1\right)\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{-255}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x \cdot \sin y, \frac{1}{z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ t_1 := z \cdot \mathsf{fma}\left(x, \frac{\sin y}{z}, 1\right)\\ \mathbf{if}\;z \leq -1.06 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{-255}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)) (t_1 (* z (fma x (/ (sin y) z) 1.0))))
   (if (<= z -1.06e+75)
     t_0
     (if (<= z -2.2e-259)
       t_1
       (if (<= z 3.95e-255) (* x (sin y)) (if (<= z 4.2e-40) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double t_1 = z * fma(x, (sin(y) / z), 1.0);
	double tmp;
	if (z <= -1.06e+75) {
		tmp = t_0;
	} else if (z <= -2.2e-259) {
		tmp = t_1;
	} else if (z <= 3.95e-255) {
		tmp = x * sin(y);
	} else if (z <= 4.2e-40) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	t_1 = Float64(z * fma(x, Float64(sin(y) / z), 1.0))
	tmp = 0.0
	if (z <= -1.06e+75)
		tmp = t_0;
	elseif (z <= -2.2e-259)
		tmp = t_1;
	elseif (z <= 3.95e-255)
		tmp = Float64(x * sin(y));
	elseif (z <= 4.2e-40)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.06e+75], t$95$0, If[LessEqual[z, -2.2e-259], t$95$1, If[LessEqual[z, 3.95e-255], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e-40], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.06e75 or 4.20000000000000036e-40 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   if -1.06e75 < z < -2.2000000000000001e-259 or 3.95000000000000005e-255 < z < 4.20000000000000036e-40

   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. 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. lower-/.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 84.1%

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

   if -2.2000000000000001e-259 < z < 3.95000000000000005e-255

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+75}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-259}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, \frac{\sin y}{z}, 1\right)\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{-255}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, \frac{\sin y}{z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -1.15e-81) t_0 (if (<= z 1.75e-41) (* x (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -1.15e-81) {
		tmp = t_0;
	} else if (z <= 1.75e-41) {
		tmp = x * sin(y);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * z
    if (z <= (-1.15d-81)) then
        tmp = t_0
    else if (z <= 1.75d-41) then
        tmp = x * sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -1.15e-81) {
		tmp = t_0;
	} else if (z <= 1.75e-41) {
		tmp = x * Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -1.15e-81:
		tmp = t_0
	elif z <= 1.75e-41:
		tmp = x * math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -1.15e-81)
		tmp = t_0;
	elseif (z <= 1.75e-41)
		tmp = Float64(x * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -1.15e-81)
		tmp = t_0;
	elseif (z <= 1.75e-41)
		tmp = x * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.15e-81], t$95$0, If[LessEqual[z, 1.75e-41], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999996e-81 or 1.75e-41 < 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 84.8

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

   5. Applied rewrites 84.8%

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

   if -1.14999999999999996e-81 < z < 1.75e-41

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-81}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -0.0215:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot -0.5, x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -0.0215)
     t_0
     (if (<= y 3.6e-22) (fma y (fma z (* y -0.5) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -0.0215) {
		tmp = t_0;
	} else if (y <= 3.6e-22) {
		tmp = fma(y, fma(z, (y * -0.5), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -0.0215)
		tmp = t_0;
	elseif (y <= 3.6e-22)
		tmp = fma(y, fma(z, Float64(y * -0.5), x), z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0215], t$95$0, If[LessEqual[y, 3.6e-22], N[(y * N[(z * N[(y * -0.5), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.021499999999999998 or 3.5999999999999998e-22 < 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 53.2

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

   5. Applied rewrites 53.2%

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

   if -0.021499999999999998 < y < 3.5999999999999998e-22

   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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 6: 40.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{+228}:\\ \;\;\;\;z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 1.06e+228) (* z 1.0) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.06e+228) {
		tmp = z * 1.0;
	} 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 <= 1.06d+228) then
        tmp = z * 1.0d0
    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 <= 1.06e+228) {
		tmp = z * 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.06e+228:
		tmp = z * 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.06e+228)
		tmp = Float64(z * 1.0);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.06e+228)
		tmp = z * 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.06e+228], N[(z * 1.0), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.06000000000000001e228

   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 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 44.4%

      \[\leadsto z \cdot 1 \]

   if 1.06000000000000001e228 < x

   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 37.8

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 37.8%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{+228}:\\ \;\;\;\;z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% 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 52.9

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

5. Applied rewrites 52.9%

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

Alternative 8: 16.4% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

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 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * x), $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 52.9

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

5. Applied rewrites 52.9%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 14.2%

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

9. Final simplification 14.2%

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

Reproduce

herbie shell --seed 2024221 
(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))))