Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-43} \lor \neg \left(x \leq 5.8 \cdot 10^{-20}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.5e-43) (not (<= x 5.8e-20)))
   (* (cos y) x)
   (fma (sin y) z (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.5e-43) || !(x <= 5.8e-20)) {
		tmp = cos(y) * x;
	} else {
		tmp = fma(sin(y), z, (1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.5e-43) || !(x <= 5.8e-20))
		tmp = Float64(cos(y) * x);
	else
		tmp = fma(sin(y), z, Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.5e-43], N[Not[LessEqual[x, 5.8e-20]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999997e-43 or 5.8e-20 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f64 57.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 57.4

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 57.4

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

   6. Applied rewrites 57.4%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 87.5

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

   9. Applied rewrites 87.5%

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

   if -3.49999999999999997e-43 < x < 5.8e-20

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 90.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-43} \lor \neg \left(x \leq 5.8 \cdot 10^{-20}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-91} \lor \neg \left(x \leq 1.2 \cdot 10^{-94}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.2e-91) (not (<= x 1.2e-94))) (* (cos y) x) (* (sin y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.2e-91) || !(x <= 1.2e-94)) {
		tmp = cos(y) * x;
	} else {
		tmp = sin(y) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.2d-91)) .or. (.not. (x <= 1.2d-94))) then
        tmp = cos(y) * x
    else
        tmp = sin(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.2e-91) || !(x <= 1.2e-94)) {
		tmp = Math.cos(y) * x;
	} else {
		tmp = Math.sin(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.2e-91) or not (x <= 1.2e-94):
		tmp = math.cos(y) * x
	else:
		tmp = math.sin(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.2e-91) || !(x <= 1.2e-94))
		tmp = Float64(cos(y) * x);
	else
		tmp = Float64(sin(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.2e-91) || ~((x <= 1.2e-94)))
		tmp = cos(y) * x;
	else
		tmp = sin(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.2e-91], N[Not[LessEqual[x, 1.2e-94]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.20000000000000048e-91 or 1.2e-94 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f64 56.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 56.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 56.8

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

   6. Applied rewrites 56.8%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 82.5

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

   9. Applied rewrites 82.5%

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

   if -8.20000000000000048e-91 < x < 1.2e-94

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 76.6

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

   5. Applied rewrites 76.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-91} \lor \neg \left(x \leq 1.2 \cdot 10^{-94}\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.02 \lor \neg \left(y \leq 0.145\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), x, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.02) (not (<= y 0.145)))
   (* (sin y) z)
   (fma
    (fma (* y y) -0.5 1.0)
    x
    (*
     (fma
      (* z (fma 0.008333333333333333 (* y y) -0.16666666666666666))
      (* y y)
      z)
     y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.02) || !(y <= 0.145)) {
		tmp = sin(y) * z;
	} else {
		tmp = fma(fma((y * y), -0.5, 1.0), x, (fma((z * fma(0.008333333333333333, (y * y), -0.16666666666666666)), (y * y), z) * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.02) || !(y <= 0.145))
		tmp = Float64(sin(y) * z);
	else
		tmp = fma(fma(Float64(y * y), -0.5, 1.0), x, Float64(fma(Float64(z * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666)), Float64(y * y), z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.02], N[Not[LessEqual[y, 0.145]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * x + N[(N[(N[(z * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0200000000000000004 or 0.14499999999999999 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 48.2

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

   5. Applied rewrites 48.2%

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

   if -0.0200000000000000004 < y < 0.14499999999999999

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right), x, \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right), x, \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{2}, 1\right), x, \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y}\right) \]

   10. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), x, \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.02 \lor \neg \left(y \leq 0.145\right):\\ \;\;\;\;\sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), x, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-73} \lor \neg \left(x \leq 1.2 \cdot 10^{-197}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e-73) (not (<= x 1.2e-197))) (* 1.0 x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e-73) || !(x <= 1.2e-197)) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	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 ((x <= (-1.15d-73)) .or. (.not. (x <= 1.2d-197))) then
        tmp = 1.0d0 * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e-73) || !(x <= 1.2e-197)) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e-73) or not (x <= 1.2e-197):
		tmp = 1.0 * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e-73) || !(x <= 1.2e-197))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.15e-73) || ~((x <= 1.2e-197)))
		tmp = 1.0 * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e-73], N[Not[LessEqual[x, 1.2e-197]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14999999999999994e-73 or 1.2e-197 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

   4. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\frac{{\left(\sin y \cdot z\right)}^{3} + {\left(\cos y \cdot x\right)}^{3}}{\mathsf{fma}\left(\sin y \cdot z - \cos y \cdot x, \sin y \cdot z, {\left(\cos y \cdot x\right)}^{2}\right)}} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 47.0%

       \[\leadsto 1 \cdot x \]

   if -1.14999999999999994e-73 < x < 1.2e-197

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-73} \lor \neg \left(x \leq 1.2 \cdot 10^{-197}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.4% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 51.2

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

5. Applied rewrites 51.2%

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

6. Final simplification 51.2%

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

Alternative 7: 17.1% accurate, 35.7× speedup

Math

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

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 * 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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 51.2

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 16.5%

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

9. Final simplification 16.5%

   \[\leadsto z \cdot y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))