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

Percentage Accurate: 99.8% → 99.8%

Time: 9.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) z (* 1.0 x))))
   (if (<= z -2.4e-45) t_0 (if (<= z 1.35e-42) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), z, (1.0 * x));
	double tmp;
	if (z <= -2.4e-45) {
		tmp = t_0;
	} else if (z <= 1.35e-42) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), z, Float64(1.0 * x))
	tmp = 0.0
	if (z <= -2.4e-45)
		tmp = t_0;
	elseif (z <= 1.35e-42)
		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 + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e-45], t$95$0, If[LessEqual[z, 1.35e-42], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3999999999999999e-45 or 1.35e-42 < z

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 89.9%

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

   if -2.3999999999999999e-45 < z < 1.35e-42

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.065:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 25500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.065)
   (* z (sin y))
   (if (<= y 25500.0)
     (fma (fma (fma -0.16666666666666666 (* z y) (* -0.5 x)) y z) y x)
     (* x (cos y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.065) {
		tmp = z * sin(y);
	} else if (y <= 25500.0) {
		tmp = fma(fma(fma(-0.16666666666666666, (z * y), (-0.5 * x)), y, z), y, x);
	} else {
		tmp = x * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.065)
		tmp = Float64(z * sin(y));
	elseif (y <= 25500.0)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(z * y), Float64(-0.5 * x)), y, z), y, x);
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.065], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 25500.0], N[(N[(N[(-0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y + z), $MachinePrecision] * y + x), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.065000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \sin y} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   if -0.065000000000000002 < y < 25500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 25500 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.065:\\ \;\;\;\;z \cdot \sin y\\ \mathbf{elif}\;y \leq 25500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 25500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.023)
     t_0
     (if (<= y 25500.0)
       (fma (fma (fma -0.16666666666666666 (* z y) (* -0.5 x)) y z) y x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.023) {
		tmp = t_0;
	} else if (y <= 25500.0) {
		tmp = fma(fma(fma(-0.16666666666666666, (z * y), (-0.5 * x)), y, z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.023)
		tmp = t_0;
	elseif (y <= 25500.0)
		tmp = fma(fma(fma(-0.16666666666666666, Float64(z * y), Float64(-0.5 * x)), y, z), y, 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, -0.023], t$95$0, If[LessEqual[y, 25500.0], N[(N[(N[(-0.16666666666666666 * N[(z * y), $MachinePrecision] + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * y + z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.023 or 25500 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if -0.023 < y < 25500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.023:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 25500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z \cdot y, -0.5 \cdot x\right), y, z\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% 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 51.6

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

5. Applied rewrites 51.6%

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

Alternative 6: 17.4% 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 51.6

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

5. Applied rewrites 51.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 15.7%

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

Reproduce

herbie shell --seed 2024243 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))