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

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto \cos y \cdot x - \sin y \cdot z \]
4. Add Preprocessing

Alternative 2: 75.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ t_1 := \frac{\sin y}{\frac{-1}{z}}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+194}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.0225:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+253}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)) (t_1 (/ (sin y) (/ -1.0 z))))
   (if (<= y -1.7e+194)
     t_0
     (if (<= y -0.0225)
       t_1
       (if (<= y 0.0055)
         (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
         (if (<= y 9.5e+122) t_0 (if (<= y 1.5e+253) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double t_1 = sin(y) / (-1.0 / z);
	double tmp;
	if (y <= -1.7e+194) {
		tmp = t_0;
	} else if (y <= -0.0225) {
		tmp = t_1;
	} else if (y <= 0.0055) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else if (y <= 9.5e+122) {
		tmp = t_0;
	} else if (y <= 1.5e+253) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	t_1 = Float64(sin(y) / Float64(-1.0 / z))
	tmp = 0.0
	if (y <= -1.7e+194)
		tmp = t_0;
	elseif (y <= -0.0225)
		tmp = t_1;
	elseif (y <= 0.0055)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	elseif (y <= 9.5e+122)
		tmp = t_0;
	elseif (y <= 1.5e+253)
		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] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e+194], t$95$0, If[LessEqual[y, -0.0225], t$95$1, If[LessEqual[y, 0.0055], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 9.5e+122], t$95$0, If[LessEqual[y, 1.5e+253], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7000000000000001e194 or 0.0054999999999999997 < y < 9.49999999999999986e122 or 1.4999999999999999e253 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -1.7000000000000001e194 < y < -0.022499999999999999 or 9.49999999999999986e122 < y < 1.4999999999999999e253

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z \cdot \sin y\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \sin y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \sin y} \]

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 26.6%

      \[\leadsto \frac{0 - {z}^{3}}{0 + \mathsf{fma}\left(z, z, 0 \cdot z\right)} \cdot \sin \color{blue}{y} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.1%

      \[\leadsto \frac{\sin y}{\color{blue}{\frac{-1}{z}}} \]

   if -0.022499999999999999 < y < 0.0054999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-z, \frac{1}{\frac{x}{\sin y}}, 1\right) \cdot x\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{+34}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma (- z) (/ 1.0 (/ x (sin y))) 1.0) x)))
   (if (<= z -1.55e-16) t_0 (if (<= z 1.62e+34) (* (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-z, (1.0 / (x / sin(y))), 1.0) * x;
	double tmp;
	if (z <= -1.55e-16) {
		tmp = t_0;
	} else if (z <= 1.62e+34) {
		tmp = cos(y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(Float64(-z), Float64(1.0 / Float64(x / sin(y))), 1.0) * x)
	tmp = 0.0
	if (z <= -1.55e-16)
		tmp = t_0;
	elseif (z <= 1.62e+34)
		tmp = Float64(cos(y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-z) * N[(1.0 / N[(x / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.55e-16], t$95$0, If[LessEqual[z, 1.62e+34], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e-16 or 1.62000000000000006e34 < z

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 87.2

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

   7. Applied rewrites 87.2%

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

   9. Applied rewrites 86.8%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 78.8%

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

   if -1.55e-16 < z < 1.62000000000000006e34

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 86.0

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

   5. Applied rewrites 86.0%

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

Alternative 4: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ t_1 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+194}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.0225:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+253}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)) (t_1 (* (- z) (sin y))))
   (if (<= y -1.7e+194)
     t_0
     (if (<= y -0.0225)
       t_1
       (if (<= y 0.0055)
         (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
         (if (<= y 6.3e+121) t_0 (if (<= y 6.6e+253) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double t_1 = -z * sin(y);
	double tmp;
	if (y <= -1.7e+194) {
		tmp = t_0;
	} else if (y <= -0.0225) {
		tmp = t_1;
	} else if (y <= 0.0055) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else if (y <= 6.3e+121) {
		tmp = t_0;
	} else if (y <= 6.6e+253) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	t_1 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (y <= -1.7e+194)
		tmp = t_0;
	elseif (y <= -0.0225)
		tmp = t_1;
	elseif (y <= 0.0055)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	elseif (y <= 6.3e+121)
		tmp = t_0;
	elseif (y <= 6.6e+253)
		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] * x), $MachinePrecision]}, Block[{t$95$1 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e+194], t$95$0, If[LessEqual[y, -0.0225], t$95$1, If[LessEqual[y, 0.0055], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 6.3e+121], t$95$0, If[LessEqual[y, 6.6e+253], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7000000000000001e194 or 0.0054999999999999997 < y < 6.29999999999999956e121 or 6.5999999999999998e253 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -1.7000000000000001e194 < y < -0.022499999999999999 or 6.29999999999999956e121 < y < 6.5999999999999998e253

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z \cdot \sin y\right)} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \sin y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \sin y} \]

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if -0.022499999999999999 < y < 0.0054999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5 \cdot x\right) \cdot y - z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -9.5e+17)
     t_0
     (if (<= y 0.0055)
       (fma (- (* (fma 0.16666666666666666 (* z y) (* -0.5 x)) y) z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -9.5e+17) {
		tmp = t_0;
	} else if (y <= 0.0055) {
		tmp = fma(((fma(0.16666666666666666, (z * y), (-0.5 * x)) * y) - z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -9.5e+17)
		tmp = t_0;
	elseif (y <= 0.0055)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), Float64(-0.5 * x)) * y) - z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -9.5e+17], t$95$0, If[LessEqual[y, 0.0055], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e17 or 0.0054999999999999997 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if -9.5e17 < y < 0.0054999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 6: 41.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-133}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-279}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e-133) (* 1.0 x) (if (<= x 2.5e-279) (* (- z) y) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-133) {
		tmp = 1.0 * x;
	} else if (x <= 2.5e-279) {
		tmp = -z * y;
	} else {
		tmp = 1.0 * 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 <= (-5d-133)) then
        tmp = 1.0d0 * x
    else if (x <= 2.5d-279) then
        tmp = -z * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-133) {
		tmp = 1.0 * x;
	} else if (x <= 2.5e-279) {
		tmp = -z * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5e-133:
		tmp = 1.0 * x
	elif x <= 2.5e-279:
		tmp = -z * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e-133)
		tmp = Float64(1.0 * x);
	elseif (x <= 2.5e-279)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5e-133)
		tmp = 1.0 * x;
	elseif (x <= 2.5e-279)
		tmp = -z * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e-133], N[(1.0 * x), $MachinePrecision], If[LessEqual[x, 2.5e-279], N[((-z) * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999999e-133 or 2.49999999999999984e-279 < x

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 94.9

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 48.1%

       \[\leadsto 1 \cdot x \]

   if -4.9999999999999999e-133 < x < 2.49999999999999984e-279

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.7%

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

Alternative 7: 52.3% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	return fma(Float64(-z), y, x)
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 54.4%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 56.3

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

7. Applied rewrites 56.3%

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

Alternative 8: 52.3% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 56.3

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

5. Applied rewrites 56.3%

   \[\leadsto \color{blue}{x - z \cdot y} \]
6. Add Preprocessing

Alternative 9: 39.4% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 * x;
}

Python

def code(x, y, z):
	return 1.0 * x

Julia

function code(x, y, z)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 54.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 93.5

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

7. Applied rewrites 93.5%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 42.1%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024257 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))