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

Percentage Accurate: 99.8% → 99.8%

Time: 12.5s

Alternatives: 7

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

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

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]

Derivation

1. Initial program 99.8%

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

Alternative 2: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ t_1 := x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+252}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))) (t_1 (* x (- 1.0 (* z (/ (sin y) x))))))
   (if (<= z -7.6e+192)
     t_0
     (if (<= z -1.75e-127)
       t_1
       (if (<= z 4.3e-114) (* x (cos y)) (if (<= z 4.1e+252) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double t_1 = x * (1.0 - (z * (sin(y) / x)));
	double tmp;
	if (z <= -7.6e+192) {
		tmp = t_0;
	} else if (z <= -1.75e-127) {
		tmp = t_1;
	} else if (z <= 4.3e-114) {
		tmp = x * cos(y);
	} else if (z <= 4.1e+252) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = sin(y) * -z
    t_1 = x * (1.0d0 - (z * (sin(y) / x)))
    if (z <= (-7.6d+192)) then
        tmp = t_0
    else if (z <= (-1.75d-127)) then
        tmp = t_1
    else if (z <= 4.3d-114) then
        tmp = x * cos(y)
    else if (z <= 4.1d+252) then
        tmp = t_1
    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.sin(y) * -z;
	double t_1 = x * (1.0 - (z * (Math.sin(y) / x)));
	double tmp;
	if (z <= -7.6e+192) {
		tmp = t_0;
	} else if (z <= -1.75e-127) {
		tmp = t_1;
	} else if (z <= 4.3e-114) {
		tmp = x * Math.cos(y);
	} else if (z <= 4.1e+252) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	t_1 = x * (1.0 - (z * (math.sin(y) / x)))
	tmp = 0
	if z <= -7.6e+192:
		tmp = t_0
	elif z <= -1.75e-127:
		tmp = t_1
	elif z <= 4.3e-114:
		tmp = x * math.cos(y)
	elif z <= 4.1e+252:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	t_1 = Float64(x * Float64(1.0 - Float64(z * Float64(sin(y) / x))))
	tmp = 0.0
	if (z <= -7.6e+192)
		tmp = t_0;
	elseif (z <= -1.75e-127)
		tmp = t_1;
	elseif (z <= 4.3e-114)
		tmp = Float64(x * cos(y));
	elseif (z <= 4.1e+252)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	t_1 = x * (1.0 - (z * (sin(y) / x)));
	tmp = 0.0;
	if (z <= -7.6e+192)
		tmp = t_0;
	elseif (z <= -1.75e-127)
		tmp = t_1;
	elseif (z <= 4.3e-114)
		tmp = x * cos(y);
	elseif (z <= 4.1e+252)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 - N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.6e+192], t$95$0, If[LessEqual[z, -1.75e-127], t$95$1, If[LessEqual[z, 4.3e-114], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.1e+252], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5999999999999999e192 or 4.1000000000000003e252 < z

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 87.9

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

   5. Simplified 87.9%

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

   if -7.5999999999999999e192 < z < -1.74999999999999995e-127 or 4.3e-114 < z < 4.1000000000000003e252

   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. flip-- N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 94.2

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 83.4%

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

   if -1.74999999999999995e-127 < z < 4.3e-114

   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 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+192}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+252}:\\ \;\;\;\;x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -0.0046:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot -0.5\right), \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= y -0.0046)
     t_0
     (if (<= y 0.5)
       (fma
        x
        (* y (* y -0.5))
        (fma (* y z) (fma 0.16666666666666666 (* y y) -1.0) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (y <= -0.0046) {
		tmp = t_0;
	} else if (y <= 0.5) {
		tmp = fma(x, (y * (y * -0.5)), fma((y * z), fma(0.16666666666666666, (y * y), -1.0), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (y <= -0.0046)
		tmp = t_0;
	elseif (y <= 0.5)
		tmp = fma(x, Float64(y * Float64(y * -0.5)), fma(Float64(y * z), fma(0.16666666666666666, Float64(y * y), -1.0), x));
	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, -0.0046], t$95$0, If[LessEqual[y, 0.5], N[(x * N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0045999999999999999 or 0.5 < y

   1. Initial program 99.6%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 59.9

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

   5. Simplified 59.9%

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

   if -0.0045999999999999999 < y < 0.5

   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 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \left(y \cdot -0.5\right), \mathsf{fma}\left(z \cdot y, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0046:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot -0.5\right), \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot -0.5\right), \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -75000000.0)
     t_0
     (if (<= y 0.036)
       (fma
        x
        (* y (* y -0.5))
        (fma (* y z) (fma 0.16666666666666666 (* y y) -1.0) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -75000000.0) {
		tmp = t_0;
	} else if (y <= 0.036) {
		tmp = fma(x, (y * (y * -0.5)), fma((y * z), fma(0.16666666666666666, (y * y), -1.0), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -75000000.0)
		tmp = t_0;
	elseif (y <= 0.036)
		tmp = fma(x, Float64(y * Float64(y * -0.5)), fma(Float64(y * z), fma(0.16666666666666666, Float64(y * y), -1.0), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -75000000.0], t$95$0, If[LessEqual[y, 0.036], N[(x * N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e7 or 0.0359999999999999973 < 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 43.7

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

   5. Simplified 43.7%

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

   if -7.5e7 < y < 0.0359999999999999973

   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 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   8. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot \left(y \cdot -0.5\right), \mathsf{fma}\left(z \cdot y, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot -0.5\right), \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot z\\ \mathbf{if}\;z \leq -7 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+259}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y z)))) (if (<= z -7e+116) t_0 (if (<= z 1.8e+259) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * z);
	double tmp;
	if (z <= -7e+116) {
		tmp = t_0;
	} else if (z <= 1.8e+259) {
		tmp = 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 = -(y * z)
    if (z <= (-7d+116)) then
        tmp = t_0
    else if (z <= 1.8d+259) then
        tmp = 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 = -(y * z);
	double tmp;
	if (z <= -7e+116) {
		tmp = t_0;
	} else if (z <= 1.8e+259) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(y * z)
	tmp = 0
	if z <= -7e+116:
		tmp = t_0
	elif z <= 1.8e+259:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * z))
	tmp = 0.0
	if (z <= -7e+116)
		tmp = t_0;
	elseif (z <= 1.8e+259)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(y * z);
	tmp = 0.0;
	if (z <= -7e+116)
		tmp = t_0;
	elseif (z <= 1.8e+259)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * z), $MachinePrecision])}, If[LessEqual[z, -7e+116], t$95$0, If[LessEqual[z, 1.8e+259], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999993e116 or 1.8000000000000001e259 < 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 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 37.3

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

   8. Simplified 37.3%

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

   if -6.99999999999999993e116 < z < 1.8000000000000001e259

   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. flip-- N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 96.2

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

   7. Simplified 96.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 49.6%

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

       1. *-rgt-identity 49.6

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

   12. Applied egg-rr 49.6%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+116}:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+259}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - N[(y * 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

   \[\leadsto \color{blue}{x + -1 \cdot \left(y \cdot z\right)} \]
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 55.1

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

5. Simplified 55.1%

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

6. Final simplification 55.1%

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

Alternative 7: 38.5% 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. 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. flip-- N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 99.6

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

4. Applied egg-rr 99.6%

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

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. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sin.f64 88.3

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

7. Simplified 88.3%

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

8. Taylor expanded in y around 0

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

10. Simplified 39.9%

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

    1. *-rgt-identity 39.9

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

12. Applied egg-rr 39.9%

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

Reproduce

herbie shell --seed 2024207 
(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))))