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

Percentage Accurate: 99.8% → 99.8%

Time: 9.9s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 87.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, x, \frac{-1}{\frac{\mathsf{fma}\left(\frac{y \cdot y}{z}, 0.16666666666666666, \frac{1}{z}\right)}{y}}\right)\\ \mathbf{if}\;x \leq -29000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (cos y)
          x
          (/ -1.0 (/ (fma (/ (* y y) z) 0.16666666666666666 (/ 1.0 z)) y)))))
   (if (<= x -29000000.0)
     t_0
     (if (<= x 1e-55) (- (* 1.0 x) (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), x, (-1.0 / (fma(((y * y) / z), 0.16666666666666666, (1.0 / z)) / y)));
	double tmp;
	if (x <= -29000000.0) {
		tmp = t_0;
	} else if (x <= 1e-55) {
		tmp = (1.0 * x) - (sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), x, Float64(-1.0 / Float64(fma(Float64(Float64(y * y) / z), 0.16666666666666666, Float64(1.0 / z)) / y)))
	tmp = 0.0
	if (x <= -29000000.0)
		tmp = t_0;
	elseif (x <= 1e-55)
		tmp = Float64(Float64(1.0 * x) - Float64(sin(y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x + N[(-1.0 / N[(N[(N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision] * 0.16666666666666666 + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -29000000.0], t$95$0, If[LessEqual[x, 1e-55], N[(N[(1.0 * x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e7 or 9.99999999999999995e-56 < 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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 89.3

          \[\leadsto \mathsf{fma}\left(\cos y, x, \frac{-1}{\frac{\mathsf{fma}\left(\frac{y \cdot y}{z}, 0.16666666666666666, \color{blue}{\frac{1}{z}}\right)}{y}}\right) \]

   9. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(\cos y, x, \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{y \cdot y}{z}, 0.16666666666666666, \frac{1}{z}\right)}{y}}}\right) \]

   if -2.9e7 < x < 9.99999999999999995e-56

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 92.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -29000000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, x, \frac{-1}{\frac{\mathsf{fma}\left(\frac{y \cdot y}{z}, 0.16666666666666666, \frac{1}{z}\right)}{y}}\right)\\ \mathbf{elif}\;x \leq 10^{-55}:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, x, \frac{-1}{\frac{\mathsf{fma}\left(\frac{y \cdot y}{z}, 0.16666666666666666, \frac{1}{z}\right)}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y - z \cdot y\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (cos y)) (* z y))))
   (if (<= x -4.9e+122)
     t_0
     (if (<= x 1.6e+160) (fma (sin y) (- z) (* 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * cos(y)) - (z * y);
	double tmp;
	if (x <= -4.9e+122) {
		tmp = t_0;
	} else if (x <= 1.6e+160) {
		tmp = fma(sin(y), -z, (1.0 * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * cos(y)) - Float64(z * y))
	tmp = 0.0
	if (x <= -4.9e+122)
		tmp = t_0;
	elseif (x <= 1.6e+160)
		tmp = fma(sin(y), Float64(-z), Float64(1.0 * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.9e+122], t$95$0, If[LessEqual[x, 1.6e+160], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8999999999999998e122 or 1.5999999999999999e160 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -4.8999999999999998e122 < x < 1.5999999999999999e160

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 83.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 5: 82.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y - z \cdot y\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (cos y)) (* z y))))
   (if (<= x -4.9e+122)
     t_0
     (if (<= x 1.6e+160) (- (* 1.0 x) (* (sin y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * cos(y)) - (z * y);
	double tmp;
	if (x <= -4.9e+122) {
		tmp = t_0;
	} else if (x <= 1.6e+160) {
		tmp = (1.0 * x) - (sin(y) * z);
	} 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 = (x * cos(y)) - (z * y)
    if (x <= (-4.9d+122)) then
        tmp = t_0
    else if (x <= 1.6d+160) then
        tmp = (1.0d0 * x) - (sin(y) * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.cos(y)) - (z * y);
	double tmp;
	if (x <= -4.9e+122) {
		tmp = t_0;
	} else if (x <= 1.6e+160) {
		tmp = (1.0 * x) - (Math.sin(y) * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.cos(y)) - (z * y)
	tmp = 0
	if x <= -4.9e+122:
		tmp = t_0
	elif x <= 1.6e+160:
		tmp = (1.0 * x) - (math.sin(y) * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * cos(y)) - Float64(z * y))
	tmp = 0.0
	if (x <= -4.9e+122)
		tmp = t_0;
	elseif (x <= 1.6e+160)
		tmp = Float64(Float64(1.0 * x) - Float64(sin(y) * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * cos(y)) - (z * y);
	tmp = 0.0;
	if (x <= -4.9e+122)
		tmp = t_0;
	elseif (x <= 1.6e+160)
		tmp = (1.0 * x) - (sin(y) * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.9e+122], t$95$0, If[LessEqual[x, 1.6e+160], N[(N[(1.0 * x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8999999999999998e122 or 1.5999999999999999e160 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -4.8999999999999998e122 < x < 1.5999999999999999e160

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 83.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+122}:\\ \;\;\;\;x \cdot \cos y - z \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;1 \cdot x - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -0.0105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 720000:\\ \;\;\;\;1 \cdot x - \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= y -0.0105)
     t_0
     (if (<= y 720000.0)
       (- (* 1.0 x) (* (* (fma -0.16666666666666666 (* y y) 1.0) z) y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (y <= -0.0105) {
		tmp = t_0;
	} else if (y <= 720000.0) {
		tmp = (1.0 * x) - ((fma(-0.16666666666666666, (y * y), 1.0) * z) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (y <= -0.0105)
		tmp = t_0;
	elseif (y <= 720000.0)
		tmp = Float64(Float64(1.0 * x) - Float64(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * z) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[y, -0.0105], t$95$0, If[LessEqual[y, 720000.0], N[(N[(1.0 * x), $MachinePrecision] - N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0105000000000000007 or 7.2e5 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   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 50.8

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

   5. Applied rewrites 50.8%

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

   if -0.0105000000000000007 < y < 7.2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.4

          \[\leadsto x \cdot 1 - \left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \cdot z\right) \cdot y \]

   8. Applied rewrites 99.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0105:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 720000:\\ \;\;\;\;1 \cdot x - \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x - \sin y \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (1.0 * x) - (math.sin(y) * z)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 * 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. Taylor expanded in y around 0

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

5. Applied rewrites 75.4%

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

6. Final simplification 75.4%

   \[\leadsto 1 \cdot x - \sin y \cdot z \]
7. Add Preprocessing

Alternative 8: 40.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-y\right) \cdot z\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y) z)))
   (if (<= z -5.8e+86) t_0 (if (<= z 3.7e+188) (fma y z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -y * z;
	double tmp;
	if (z <= -5.8e+86) {
		tmp = t_0;
	} else if (z <= 3.7e+188) {
		tmp = fma(y, z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-y) * z)
	tmp = 0.0
	if (z <= -5.8e+86)
		tmp = t_0;
	elseif (z <= 3.7e+188)
		tmp = fma(y, z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) * z), $MachinePrecision]}, If[LessEqual[z, -5.8e+86], t$95$0, If[LessEqual[z, 3.7e+188], N[(y * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.79999999999999981e86 or 3.7e188 < z

   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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.3%

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

   if -5.79999999999999981e86 < z < 3.7e188

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 44.9%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 44.6

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

   10. Applied rewrites 44.6%

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

Alternative 9: 51.9% 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 49.3

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

5. Applied rewrites 49.3%

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

Alternative 10: 37.5% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 75.4%

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

7. Applied rewrites 35.4%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 35.4

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

10. Applied rewrites 35.4%

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

Reproduce

herbie shell --seed 2024295 
(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))))