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

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(x * N[Cos[y], $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. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 99.9

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

5. Simplified 99.9%

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

Alternative 2: 85.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, \sin y, x\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (sin y) x)))
   (if (<= z -1.7e+14) t_0 (if (<= z 5.6e-32) (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, sin(y), x);
	double tmp;
	if (z <= -1.7e+14) {
		tmp = t_0;
	} else if (z <= 5.6e-32) {
		tmp = x * cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, sin(y), x)
	tmp = 0.0
	if (z <= -1.7e+14)
		tmp = t_0;
	elseif (z <= 5.6e-32)
		tmp = Float64(x * cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.7e+14], t$95$0, If[LessEqual[z, 5.6e-32], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e14 or 5.5999999999999998e-32 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 91.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sin.f64 91.0

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

   11. Simplified 91.0%

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

   if -1.7e14 < z < 5.5999999999999998e-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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= z -9.2e+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 = z * sin(y);
	double tmp;
	if (z <= -9.2e+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 = z * sin(y)
    if (z <= (-9.2d+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 = z * Math.sin(y);
	double tmp;
	if (z <= -9.2e+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 = z * math.sin(y)
	tmp = 0
	if z <= -9.2e+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(z * sin(y))
	tmp = 0.0
	if (z <= -9.2e+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 = z * sin(y);
	tmp = 0.0;
	if (z <= -9.2e+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[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+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 < -9.2000000000000006e141 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

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 73.1

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

   5. Simplified 73.1%

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

   if -9.2000000000000006e141 < 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 80.2

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

   5. Simplified 80.2%

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

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos y\\ \mathbf{if}\;y \leq -0.16:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.096:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (cos y))))
   (if (<= y -0.16)
     t_0
     (if (<= y 0.096)
       (fma
        x
        (fma y (* y (fma (* y y) 0.041666666666666664 -0.5)) 1.0)
        (*
         z
         (fma
          (* y (fma y (* y 0.008333333333333333) -0.16666666666666666))
          (* y y)
          y)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * cos(y);
	double tmp;
	if (y <= -0.16) {
		tmp = t_0;
	} else if (y <= 0.096) {
		tmp = fma(x, fma(y, (y * fma((y * y), 0.041666666666666664, -0.5)), 1.0), (z * fma((y * fma(y, (y * 0.008333333333333333), -0.16666666666666666)), (y * y), y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * cos(y))
	tmp = 0.0
	if (y <= -0.16)
		tmp = t_0;
	elseif (y <= 0.096)
		tmp = fma(x, fma(y, Float64(y * fma(Float64(y * y), 0.041666666666666664, -0.5)), 1.0), Float64(z * fma(Float64(y * fma(y, Float64(y * 0.008333333333333333), -0.16666666666666666)), Float64(y * y), y)));
	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, -0.16], t$95$0, If[LessEqual[y, 0.096], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(z * N[(N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.160000000000000003 or 0.096000000000000002 < 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 53.2

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

   5. Simplified 53.2%

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

   if -0.160000000000000003 < y < 0.096000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.041666666666666664, -0.5\right), 1\right), z \cdot \sin y\right) \]

   8. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right)}, z \cdot \sin y\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + y \cdot 1\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2}\right)} + y \cdot 1\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \left(\color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot {y}^{2}} + y \cdot 1\right)\right) \]

       5. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \left(\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot {y}^{2} + \color{blue}{y}\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right), {y}^{2}, y\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, {y}^{2}, y\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, {y}^{2}, y\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), {y}^{2}, y\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), {y}^{2}, y\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), {y}^{2}, y\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}\right), {y}^{2}, y\right)\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right)}, {y}^{2}, y\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{-1}{6}\right), {y}^{2}, y\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{y \cdot y}, y\right)\right) \]

       16. lower-*.f64 99.7

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z \cdot \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), \color{blue}{y \cdot y}, y\right)\right) \]

   11. Simplified 99.7%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z \cdot \color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), y \cdot y, y\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.2% 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 54.9

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

5. Simplified 54.9%

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

Alternative 6: 17.2% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lower-*.f64 N/A

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

   2. lower-sin.f64 35.8

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

5. Simplified 35.8%

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

6. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 15.0

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

8. Simplified 15.0%

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

9. Final simplification 15.0%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))