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

Percentage Accurate: 99.8% → 99.8%

Time: 10.4s

Alternatives: 6

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: 85.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -z, x\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) (- z) x)))
   (if (<= z -1.7e+14) t_0 (if (<= z 6e-32) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), -z, x);
	double tmp;
	if (z <= -1.7e+14) {
		tmp = t_0;
	} else if (z <= 6e-32) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), Float64(-z), x)
	tmp = 0.0
	if (z <= -1.7e+14)
		tmp = t_0;
	elseif (z <= 6e-32)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z) + x), $MachinePrecision]}, If[LessEqual[z, -1.7e+14], t$95$0, If[LessEqual[z, 6e-32], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e14 or 6.0000000000000001e-32 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 91.0%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-neg.f64 91.0

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

   8. Simplified 91.0%

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

   if -1.7e14 < z < 6.0000000000000001e-32

   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 87.0

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

   5. Simplified 87.0%

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

Alternative 3: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= z -7.8e+141) t_0 (if (<= z 6.8e+86) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (z <= -7.8e+141) {
		tmp = t_0;
	} else if (z <= 6.8e+86) {
		tmp = x * cos(y);
	} 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 = sin(y) * -z
    if (z <= (-7.8d+141)) then
        tmp = t_0
    else if (z <= 6.8d+86) then
        tmp = x * cos(y)
    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 tmp;
	if (z <= -7.8e+141) {
		tmp = t_0;
	} else if (z <= 6.8e+86) {
		tmp = x * Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	tmp = 0
	if z <= -7.8e+141:
		tmp = t_0
	elif z <= 6.8e+86:
		tmp = x * math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (z <= -7.8e+141)
		tmp = t_0;
	elseif (z <= 6.8e+86)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	tmp = 0.0;
	if (z <= -7.8e+141)
		tmp = t_0;
	elseif (z <= 6.8e+86)
		tmp = x * cos(y);
	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]}, If[LessEqual[z, -7.8e+141], t$95$0, If[LessEqual[z, 6.8e+86], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.79999999999999983e141 or 6.7999999999999995e86 < 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 73.2

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

   5. Simplified 73.2%

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

   if -7.79999999999999983e141 < z < 6.7999999999999995e86

   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 80.4

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

   5. Simplified 80.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+141}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.078:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.15)
     t_0
     (if (<= y 0.078)
       (fma
        y
        (* z (fma y (* y 0.16666666666666666) -1.0))
        (fma (* x (* y y)) (fma (* y y) 0.041666666666666664 -0.5) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.15) {
		tmp = t_0;
	} else if (y <= 0.078) {
		tmp = fma(y, (z * fma(y, (y * 0.16666666666666666), -1.0)), fma((x * (y * y)), fma((y * y), 0.041666666666666664, -0.5), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_0;
	elseif (y <= 0.078)
		tmp = fma(y, Float64(z * fma(y, Float64(y * 0.16666666666666666), -1.0)), fma(Float64(x * Float64(y * y)), fma(Float64(y * y), 0.041666666666666664, -0.5), 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, -0.15], t$95$0, If[LessEqual[y, 0.078], N[(y * N[(z * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 0.0779999999999999999 < y

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   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 53.4

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

   5. Simplified 53.4%

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

   if -0.149999999999999994 < y < 0.0779999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + {y}^{2} \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot x + \frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right) + x\right)} - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      2. +-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(x \cdot {y}^{2}\right) + \frac{-1}{2} \cdot x\right)} + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{24} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{-1}{2} \cdot x\right) \cdot {y}^{2}\right)} + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(x \cdot {y}^{2}\right) \cdot \frac{1}{24}\right)} \cdot {y}^{2} + \left(\frac{-1}{2} \cdot x\right) \cdot {y}^{2}\right) + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\left(\color{blue}{\left(x \cdot {y}^{2}\right) \cdot \left(\frac{1}{24} \cdot {y}^{2}\right)} + \left(\frac{-1}{2} \cdot x\right) \cdot {y}^{2}\right) + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(x \cdot {y}^{2}\right) \cdot \left(\frac{1}{24} \cdot {y}^{2}\right) + \color{blue}{\frac{-1}{2} \cdot \left(x \cdot {y}^{2}\right)}\right) + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot {y}^{2}\right) \cdot \left(\frac{1}{24} \cdot {y}^{2}\right) + \color{blue}{\left(x \cdot {y}^{2}\right) \cdot \frac{-1}{2}}\right) + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot {y}^{2}\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}\right)} + x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {y}^{2}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right)} - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot x}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot x, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot x\right)}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(x \cdot y\right)}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(x \cdot y\right)}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(y \cdot x\right)}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(y \cdot x\right)}, \frac{1}{24} \cdot {y}^{2} + \frac{-1}{2}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot x\right), \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {y}^{2}, \frac{-1}{2}\right)}, x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot x\right), \mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \frac{-1}{6}\right), y\right) \]

      19. lower-*.f64 99.6

          \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot x\right), \mathsf{fma}\left(0.041666666666666664, \color{blue}{y \cdot y}, -0.5\right), x\right) - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot -0.16666666666666666\right), y\right) \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot x\right), \mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), x\right)} - z \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot -0.16666666666666666\right), y\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right) - z \cdot \left(y + \frac{-1}{6} \cdot {y}^{3}\right)} \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right), \mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 0.078:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.2% 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 54.8

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

5. Simplified 54.8%

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

6. Final simplification 54.8%

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

Alternative 6: 17.2% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(-y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z * (-y)), $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 54.8

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

5. Simplified 54.8%

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

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 14.9

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

8. Simplified 14.9%

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

Reproduce

herbie shell --seed 2024215 
(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))))