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

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+135} \lor \neg \left(z \leq 8.5 \cdot 10^{+67}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.1e+135) (not (<= z 8.5e+67)))
   (* (cos y) z)
   (+ (* x (sin y)) (* z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e+135) || !(z <= 8.5e+67)) {
		tmp = cos(y) * z;
	} else {
		tmp = (x * sin(y)) + (z * 1.0);
	}
	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) :: tmp
    if ((z <= (-2.1d+135)) .or. (.not. (z <= 8.5d+67))) then
        tmp = cos(y) * z
    else
        tmp = (x * sin(y)) + (z * 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e+135) || !(z <= 8.5e+67)) {
		tmp = Math.cos(y) * z;
	} else {
		tmp = (x * Math.sin(y)) + (z * 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.1e+135) or not (z <= 8.5e+67):
		tmp = math.cos(y) * z
	else:
		tmp = (x * math.sin(y)) + (z * 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.1e+135) || !(z <= 8.5e+67))
		tmp = Float64(cos(y) * z);
	else
		tmp = Float64(Float64(x * sin(y)) + Float64(z * 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.1e+135) || ~((z <= 8.5e+67)))
		tmp = cos(y) * z;
	else
		tmp = (x * sin(y)) + (z * 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.1e+135], N[Not[LessEqual[z, 8.5e+67]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e135 or 8.50000000000000038e67 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -2.1000000000000001e135 < z < 8.50000000000000038e67

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 87.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+135} \lor \neg \left(z \leq 8.5 \cdot 10^{+67}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sin y + z \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-60} \lor \neg \left(z \leq 2.5 \cdot 10^{-62}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e-60) (not (<= z 2.5e-62))) (* (cos y) z) (* (sin y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-60) || !(z <= 2.5e-62)) {
		tmp = cos(y) * z;
	} else {
		tmp = sin(y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.1d-60)) .or. (.not. (z <= 2.5d-62))) then
        tmp = cos(y) * z
    else
        tmp = sin(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-60) || !(z <= 2.5e-62)) {
		tmp = Math.cos(y) * z;
	} else {
		tmp = Math.sin(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e-60) or not (z <= 2.5e-62):
		tmp = math.cos(y) * z
	else:
		tmp = math.sin(y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e-60) || !(z <= 2.5e-62))
		tmp = Float64(cos(y) * z);
	else
		tmp = Float64(sin(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.1e-60) || ~((z <= 2.5e-62)))
		tmp = cos(y) * z;
	else
		tmp = sin(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e-60], N[Not[LessEqual[z, 2.5e-62]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0999999999999999e-60 or 2.5000000000000001e-62 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -1.0999999999999999e-60 < z < 2.5000000000000001e-62

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 56.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 56.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 56.7

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\sin y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sin y \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin y \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sin y}\right)\right) \cdot x \]

      5. unpow2 N/A

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

      6. rem-square-sqrt N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 76.1

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

   7. Applied rewrites 76.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-60} \lor \neg \left(z \leq 2.5 \cdot 10^{-62}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15200 \lor \neg \left(y \leq 10.2\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -15200.0) (not (<= y 10.2)))
   (* (cos y) z)
   (fma (fma (fma -0.16666666666666666 (* y x) (* -0.5 z)) y x) y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -15200.0) || !(y <= 10.2)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, (y * x), (-0.5 * z)), y, x), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -15200.0) || !(y <= 10.2))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(fma(fma(-0.16666666666666666, Float64(y * x), Float64(-0.5 * z)), y, x), y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -15200.0], N[Not[LessEqual[y, 10.2]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(y * x), $MachinePrecision] + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -15200 or 10.199999999999999 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.6

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

   5. Applied rewrites 52.6%

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

   if -15200 < y < 10.199999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15200 \lor \neg \left(y \leq 10.2\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot x, -0.5 \cdot z\right), y, x\right), y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 51.7

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

5. Applied rewrites 51.7%

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

Alternative 6: 17.2% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 51.7

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 13.8%

   \[\leadsto x \cdot \color{blue}{y} \]
9. Add Preprocessing

Alternative 7: 4.8% accurate, 71.3× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow1 N/A

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

   4. metadata-eval N/A

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

   5. sqrt-pow1 N/A

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

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lift-*.f64 N/A

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

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

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. sqrt-prod N/A

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

   15. lower-fma.f64 N/A

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

4. Applied rewrites 24.0%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot z} \]

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 4.2

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

7. Applied rewrites 4.2%

   \[\leadsto \color{blue}{-z} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))