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

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

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: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{-143} \lor \neg \left(x \leq 4.4 \cdot 10^{-74}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.9e-143) (not (<= x 4.4e-74)))
   (fma (sin y) x (* 1.0 z))
   (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.9e-143) || !(x <= 4.4e-74)) {
		tmp = fma(sin(y), x, (1.0 * z));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.9e-143) || !(x <= 4.4e-74))
		tmp = fma(sin(y), x, Float64(1.0 * z));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.9e-143], N[Not[LessEqual[x, 4.4e-74]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * x + N[(1.0 * z), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.89999999999999989e-143 or 4.40000000000000021e-74 < x

   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 86.5%

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

   if -6.89999999999999989e-143 < x < 4.40000000000000021e-74

   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 90.6

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

   5. Applied rewrites 90.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{-143} \lor \neg \left(x \leq 4.4 \cdot 10^{-74}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin y, x, 1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+103} \lor \neg \left(x \leq 7.2 \cdot 10^{+108}\right):\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6e+103) (not (<= x 7.2e+108))) (* (sin y) x) (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e+103) || !(x <= 7.2e+108)) {
		tmp = sin(y) * x;
	} else {
		tmp = cos(y) * z;
	}
	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 <= (-1.6d+103)) .or. (.not. (x <= 7.2d+108))) then
        tmp = sin(y) * x
    else
        tmp = cos(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e+103) || !(x <= 7.2e+108)) {
		tmp = Math.sin(y) * x;
	} else {
		tmp = Math.cos(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.6e+103) or not (x <= 7.2e+108):
		tmp = math.sin(y) * x
	else:
		tmp = math.cos(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6e+103) || !(x <= 7.2e+108))
		tmp = Float64(sin(y) * x);
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.6e+103) || ~((x <= 7.2e+108)))
		tmp = sin(y) * x;
	else
		tmp = cos(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.6e+103], N[Not[LessEqual[x, 7.2e+108]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999996e103 or 7.2e108 < x

   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. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot x, \frac{x}{\mathsf{fma}\left(-z, \cos y, \sin y \cdot x\right)}, -\frac{{\left(\cos y \cdot z\right)}^{2}}{\mathsf{fma}\left(-z, \cos y, \sin y \cdot x\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-neg-frac2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 91.7%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 79.8

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

   9. Applied rewrites 79.8%

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

   if -1.59999999999999996e103 < x < 7.2e108

   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 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+103} \lor \neg \left(x \leq 7.2 \cdot 10^{+108}\right):\\ \;\;\;\;\sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000335 \lor \neg \left(y \leq 0.028\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot y\right), y, x\right), y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.000335) (not (<= y 0.028)))
   (* (cos y) z)
   (fma (fma (* -0.16666666666666666 (* x y)) y x) y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.000335) || !(y <= 0.028)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(fma((-0.16666666666666666 * (x * y)), y, x), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.000335) || !(y <= 0.028))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(fma(Float64(-0.16666666666666666 * Float64(x * y)), y, x), y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.000335], N[Not[LessEqual[y, 0.028]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3500000000000001e-4 or 0.0280000000000000006 < 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 49.3

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

   5. Applied rewrites 49.3%

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

   if -3.3500000000000001e-4 < y < 0.0280000000000000006

   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. 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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000335 \lor \neg \left(y \leq 0.028\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot y\right), y, x\right), y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-205} \lor \neg \left(z \leq 3.2 \cdot 10^{-184}\right):\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.6e-205) (not (<= z 3.2e-184))) (* 1.0 z) (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.6e-205) || !(z <= 3.2e-184)) {
		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 <= (-1.6d-205)) .or. (.not. (z <= 3.2d-184))) 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 <= -1.6e-205) || !(z <= 3.2e-184)) {
		tmp = 1.0 * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.6e-205) or not (z <= 3.2e-184):
		tmp = 1.0 * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.6e-205) || !(z <= 3.2e-184))
		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 <= -1.6e-205) || ~((z <= 3.2e-184)))
		tmp = 1.0 * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.6e-205], N[Not[LessEqual[z, 3.2e-184]], $MachinePrecision]], N[(1.0 * z), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000005e-205 or 3.2e-184 < 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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 44.2%

      \[\leadsto 1 \cdot z \]

   if -1.60000000000000005e-205 < z < 3.2e-184

   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 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 35.4%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-205} \lor \neg \left(z \leq 3.2 \cdot 10^{-184}\right):\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% 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 49.2

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

5. Applied rewrites 49.2%

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

6. Final simplification 49.2%

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

Alternative 7: 17.3% 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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 14.1%

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

9. Final simplification 14.1%

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

Reproduce

herbie shell --seed 2024314 
(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))))