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

Percentage Accurate: 99.8% → 99.8%

Time: 4.9s

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

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

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-+.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. Final simplification 99.8%

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

Alternative 2: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+82}:\\ \;\;\;\;1 \cdot z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -7.8e+18)
     t_0
     (if (<= z 7.8e+82) (+ (* 1.0 z) (* x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -7.8e+18) {
		tmp = t_0;
	} else if (z <= 7.8e+82) {
		tmp = (1.0 * z) + (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 = z * cos(y)
    if (z <= (-7.8d+18)) then
        tmp = t_0
    else if (z <= 7.8d+82) then
        tmp = (1.0d0 * z) + (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 = z * Math.cos(y);
	double tmp;
	if (z <= -7.8e+18) {
		tmp = t_0;
	} else if (z <= 7.8e+82) {
		tmp = (1.0 * z) + (x * Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -7.8e+18:
		tmp = t_0
	elif z <= 7.8e+82:
		tmp = (1.0 * z) + (x * math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -7.8e+18)
		tmp = t_0;
	elseif (z <= 7.8e+82)
		tmp = Float64(Float64(1.0 * z) + Float64(x * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -7.8e+18)
		tmp = t_0;
	elseif (z <= 7.8e+82)
		tmp = (1.0 * z) + (x * sin(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e+18], t$95$0, If[LessEqual[z, 7.8e+82], N[(N[(1.0 * z), $MachinePrecision] + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8e18 or 7.79999999999999951e82 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   if -7.8e18 < z < 7.79999999999999951e82

   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 91.0%

      \[\leadsto x \cdot \sin y + z \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+82}:\\ \;\;\;\;1 \cdot z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.84:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) x)))
   (if (<= y -1.35e+171)
     t_0
     (if (<= y -0.84)
       (* z (cos y))
       (if (<= y 1.45e-11)
         (fma (* (fma -0.16666666666666666 (* y y) 1.0) x) y z)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * x;
	double tmp;
	if (y <= -1.35e+171) {
		tmp = t_0;
	} else if (y <= -0.84) {
		tmp = z * cos(y);
	} else if (y <= 1.45e-11) {
		tmp = fma((fma(-0.16666666666666666, (y * y), 1.0) * x), 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 <= -1.35e+171)
		tmp = t_0;
	elseif (y <= -0.84)
		tmp = Float64(z * cos(y));
	elseif (y <= 1.45e-11)
		tmp = fma(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * x), 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, -1.35e+171], t$95$0, If[LessEqual[y, -0.84], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e-11], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * y + z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3499999999999999e171 or 1.45e-11 < 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. 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 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 91.7

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

   7. Applied rewrites 91.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \sin y \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 63.3%

       \[\leadsto \sin y \cdot x \]

   if -1.3499999999999999e171 < y < -0.839999999999999969

   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. *-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 61.1

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

   5. Applied rewrites 61.1%

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

   if -0.839999999999999969 < y < 1.45e-11

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right)} \]
   6. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right), y, z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot x, y, z\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+171}:\\ \;\;\;\;\sin y \cdot x\\ \mathbf{elif}\;y \leq -0.84:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -7.8e+18) t_0 (if (<= z 7.8e+82) (fma (sin y) x (* 1.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -7.8e+18) {
		tmp = t_0;
	} else if (z <= 7.8e+82) {
		tmp = fma(sin(y), x, (1.0 * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -7.8e+18)
		tmp = t_0;
	elseif (z <= 7.8e+82)
		tmp = fma(sin(y), x, Float64(1.0 * 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[z, -7.8e+18], t$95$0, If[LessEqual[z, 7.8e+82], N[(N[Sin[y], $MachinePrecision] * x + N[(1.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8e18 or 7.79999999999999951e82 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   if -7.8e18 < z < 7.79999999999999951e82

   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. Taylor expanded in y around 0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= y -0.84)
     t_0
     (if (<= y 4.7e+14)
       (fma (* (fma -0.16666666666666666 (* y y) 1.0) x) y z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (y <= -0.84) {
		tmp = t_0;
	} else if (y <= 4.7e+14) {
		tmp = fma((fma(-0.16666666666666666, (y * y), 1.0) * x), y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -0.84)
		tmp = t_0;
	elseif (y <= 4.7e+14)
		tmp = fma(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * x), y, 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, -0.84], t$95$0, If[LessEqual[y, 4.7e+14], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * y + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.839999999999999969 or 4.7e14 < 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. *-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 45.3

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

   5. Applied rewrites 45.3%

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

   if -0.839999999999999969 < y < 4.7e14

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + y \cdot \left(x + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot \left(x \cdot y\right)\right)\right)} \]
   6. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right), y, z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot x, y, z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+184}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+215}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.95e+184) (* x y) (if (<= x 2.6e+215) (* 1.0 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e+184) {
		tmp = x * y;
	} else if (x <= 2.6e+215) {
		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 (x <= (-1.95d+184)) then
        tmp = x * y
    else if (x <= 2.6d+215) 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 (x <= -1.95e+184) {
		tmp = x * y;
	} else if (x <= 2.6e+215) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.95e+184:
		tmp = x * y
	elif x <= 2.6e+215:
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.95e+184)
		tmp = Float64(x * y);
	elseif (x <= 2.6e+215)
		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 (x <= -1.95e+184)
		tmp = x * y;
	elseif (x <= 2.6e+215)
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.95e+184], N[(x * y), $MachinePrecision], If[LessEqual[x, 2.6e+215], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999985e184 or 2.6e215 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + x \cdot y} \]
   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 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.0%

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

   if -1.94999999999999985e184 < x < 2.6e215

   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 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.4%

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

Alternative 7: 52.2% 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 53.7

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

5. Applied rewrites 53.7%

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

Alternative 8: 16.1% 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 53.7

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

5. Applied rewrites 53.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 16.1%

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

Reproduce

herbie shell --seed 2024332 
(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))))