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

Percentage Accurate: 99.8% → 99.8%

Time: 16.0s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;x \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\frac{1}{z}}, \sin y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= x -1e+80)
     t_0
     (if (<= x 6.2e-78) (fma (/ -1.0 (/ 1.0 z)) (sin y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (x <= -1e+80) {
		tmp = t_0;
	} else if (x <= 6.2e-78) {
		tmp = fma((-1.0 / (1.0 / z)), sin(y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (x <= -1e+80)
		tmp = t_0;
	elseif (x <= 6.2e-78)
		tmp = fma(Float64(-1.0 / Float64(1.0 / z)), sin(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[x, -1e+80], t$95$0, If[LessEqual[x, 6.2e-78], N[(N[(-1.0 / N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e80 or 6.20000000000000035e-78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 89.7

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

   5. Simplified 89.7%

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

   if -1e80 < x < 6.20000000000000035e-78

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - z \cdot \sin y\right)} + x \cdot \cos y \]

      8. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{0 + z \cdot \sin y}} + x \cdot \cos y \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{0 + z \cdot \sin y} + x \cdot \cos y \]

      10. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right)}}{0 + z \cdot \sin y} + x \cdot \cos y \]

      11. div-inv N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right)\right) \cdot \frac{1}{0 + z \cdot \sin y}} + x \cdot \cos y \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)\right), \frac{1}{0 + z \cdot \sin y}, x \cdot \cos y\right)} \]

   4. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-z \cdot z\right), \frac{1}{0 + z \cdot \sin y}, x \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot z\right)\right), \frac{1}{0 + z \cdot \sin y}, x \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 39.7%

      \[\leadsto \mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-z \cdot z\right), \frac{1}{0 + z \cdot \sin y}, x \cdot \color{blue}{1}\right) \]

   9. Applied egg-rr 91.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\frac{1}{z}}, \sin y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;x \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;x - x \cdot \left(z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= x -1e+80)
     t_0
     (if (<= x 6.2e-78) (- x (* x (* z (/ (sin y) x)))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (x <= -1e+80) {
		tmp = t_0;
	} else if (x <= 6.2e-78) {
		tmp = x - (x * (z * (sin(y) / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * x
    if (x <= (-1d+80)) then
        tmp = t_0
    else if (x <= 6.2d-78) then
        tmp = x - (x * (z * (sin(y) / x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * x;
	double tmp;
	if (x <= -1e+80) {
		tmp = t_0;
	} else if (x <= 6.2e-78) {
		tmp = x - (x * (z * (Math.sin(y) / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * x
	tmp = 0
	if x <= -1e+80:
		tmp = t_0
	elif x <= 6.2e-78:
		tmp = x - (x * (z * (math.sin(y) / x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (x <= -1e+80)
		tmp = t_0;
	elseif (x <= 6.2e-78)
		tmp = Float64(x - Float64(x * Float64(z * Float64(sin(y) / x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * x;
	tmp = 0.0;
	if (x <= -1e+80)
		tmp = t_0;
	elseif (x <= 6.2e-78)
		tmp = x - (x * (z * (sin(y) / x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1e+80], t$95$0, If[LessEqual[x, 6.2e-78], N[(x - N[(x * N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e80 or 6.20000000000000035e-78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 89.7

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

   5. Simplified 89.7%

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

   if -1e80 < x < 6.20000000000000035e-78

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 91.6%

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

   10. Taylor expanded in x around inf

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. mul-1-neg N/A

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

       4. distribute-lft-neg-out N/A

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

       5. unsub-neg N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. associate-/l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-sin.f64 75.0

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

   12. Simplified 75.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;x - x \cdot \left(z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -z \cdot \sin y\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+265}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.075:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* z (sin y)))))
   (if (<= y -4.3e+265)
     t_0
     (if (<= y -0.075)
       (* (cos y) x)
       (if (<= y 0.1)
         (fma y (fma y (fma z (* y 0.16666666666666666) (* x -0.5)) (- z)) x)
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -(z * sin(y));
	double tmp;
	if (y <= -4.3e+265) {
		tmp = t_0;
	} else if (y <= -0.075) {
		tmp = cos(y) * x;
	} else if (y <= 0.1) {
		tmp = fma(y, fma(y, fma(z, (y * 0.16666666666666666), (x * -0.5)), -z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(z * sin(y)))
	tmp = 0.0
	if (y <= -4.3e+265)
		tmp = t_0;
	elseif (y <= -0.075)
		tmp = Float64(cos(y) * x);
	elseif (y <= 0.1)
		tmp = fma(y, fma(y, fma(z, Float64(y * 0.16666666666666666), Float64(x * -0.5)), Float64(-z)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[y, -4.3e+265], t$95$0, If[LessEqual[y, -0.075], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 0.1], N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.29999999999999973e265 or 0.10000000000000001 < 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 59.8

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

   5. Simplified 59.8%

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

   if -4.29999999999999973e265 < y < -0.0749999999999999972

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 63.5

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

   5. Simplified 63.5%

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

   if -0.0749999999999999972 < y < 0.10000000000000001

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+265}:\\ \;\;\;\;-z \cdot \sin y\\ \mathbf{elif}\;y \leq -0.075:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.075:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 25:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) x)))
   (if (<= y -0.075)
     t_0
     (if (<= y 25.0)
       (fma y (fma y (fma z (* y 0.16666666666666666) (* x -0.5)) (- z)) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * x;
	double tmp;
	if (y <= -0.075) {
		tmp = t_0;
	} else if (y <= 25.0) {
		tmp = fma(y, fma(y, fma(z, (y * 0.16666666666666666), (x * -0.5)), -z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * x)
	tmp = 0.0
	if (y <= -0.075)
		tmp = t_0;
	elseif (y <= 25.0)
		tmp = fma(y, fma(y, fma(z, Float64(y * 0.16666666666666666), Float64(x * -0.5)), Float64(-z)), 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, -0.075], t$95$0, If[LessEqual[y, 25.0], N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0749999999999999972 or 25 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 52.8

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

   5. Simplified 52.8%

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

   if -0.0749999999999999972 < y < 25

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.075:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq 25:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, x \cdot -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, x, \frac{-1}{\frac{\mathsf{fma}\left(0.16666666666666666, \frac{y \cdot y}{z}, \frac{1}{z}\right)}{y}}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma 1.0 x (/ -1.0 (/ (fma 0.16666666666666666 (/ (* y y) z) (/ 1.0 z)) y))))

C

double code(double x, double y, double z) {
	return fma(1.0, x, (-1.0 / (fma(0.16666666666666666, ((y * y) / z), (1.0 / z)) / y)));
}

Julia

function code(x, y, z)
	return fma(1.0, x, Float64(-1.0 / Float64(fma(0.16666666666666666, Float64(Float64(y * y) / z), Float64(1.0 / z)) / y)))
end

Wolfram

code[x_, y_, z_] := N[(1.0 * x + N[(-1.0 / N[(N[(0.16666666666666666 * N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision] + N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.8

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in y around 0

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

9. Simplified 77.5%

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

10. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. associate-*r/ N/A

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

    6. lower-/.f64 N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. associate-*r/ N/A

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

    10. associate-*l/ N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. unpow2 N/A

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

    15. lower-*.f64 N/A

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

    16. lower-/.f64 56.1

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

12. Simplified 56.1%

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

Alternative 8: 53.0% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+28}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y -5.2e+28) (- x) (- x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+28) {
		tmp = -x;
	} else {
		tmp = x - (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 (y <= (-5.2d+28)) then
        tmp = -x
    else
        tmp = x - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+28) {
		tmp = -x;
	} else {
		tmp = x - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.2e+28:
		tmp = -x
	else:
		tmp = x - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.2e+28)
		tmp = Float64(-x);
	else
		tmp = Float64(x - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.2e+28)
		tmp = -x;
	else
		tmp = x - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.2e+28], (-x), N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000004e28

   1. Initial program 99.5%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 4.2

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

   5. Simplified 4.2%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\mathsf{neg}\left(z \cdot y\right)\right)}^{3}}{x \cdot x + \left(\left(\mathsf{neg}\left(z \cdot y\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right) - x \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right)\right)}} \]

      4. cube-neg N/A

         \[\leadsto \frac{{x}^{3} + \color{blue}{\left(\mathsf{neg}\left({\left(z \cdot y\right)}^{3}\right)\right)}}{x \cdot x + \left(\left(\mathsf{neg}\left(z \cdot y\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right) - x \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{\color{blue}{{x}^{3} - {\left(z \cdot y\right)}^{3}}}{x \cdot x + \left(\left(\mathsf{neg}\left(z \cdot y\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right) - x \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(z \cdot y\right)}^{3}}{x \cdot x + \left(\left(\mathsf{neg}\left(z \cdot y\right)\right) \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right) - x \cdot \left(\mathsf{neg}\left(z \cdot y\right)\right)\right)}} \]

   7. Applied egg-rr 1.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 2.7

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

   10. Simplified 2.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{-1 \cdot x} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

       2. lower-neg.f64 11.0

          \[\leadsto \color{blue}{-x} \]

   13. Simplified 11.0%

       \[\leadsto \color{blue}{-x} \]

   if -5.2000000000000004e28 < y

   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 71.1

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

   5. Simplified 71.1%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+28}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+241}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.15e+241) x (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.15e+241) {
		tmp = x;
	} else {
		tmp = 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 (z <= 1.15d+241) then
        tmp = x
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.15e+241) {
		tmp = x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.15e+241:
		tmp = x
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.15e+241)
		tmp = x;
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.15e+241)
		tmp = x;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.15e+241], x, N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e241

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 67.7

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

   5. Simplified 67.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 44.3%

      \[\leadsto x \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 44.3

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

   10. Applied egg-rr 44.3%

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

   if 1.15e241 < z

   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 66.2

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

   5. Simplified 66.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 49.5

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

   8. Simplified 49.5%

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

Alternative 10: 38.9% accurate, 214.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 64.5

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

5. Simplified 64.5%

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

6. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 42.8%

   \[\leadsto x \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 42.8

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

10. Applied egg-rr 42.8%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024211 
(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))))