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

Percentage Accurate: 99.8% → 99.8%

Time: 11.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + N[(x * N[Sin[y], $MachinePrecision]), $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-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 1.0 z (* x (sin y)))))
   (if (<= x -6e-61) t_0 (if (<= x 2.7e-17) (* (cos y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(1.0, z, (x * sin(y)));
	double tmp;
	if (x <= -6e-61) {
		tmp = t_0;
	} else if (x <= 2.7e-17) {
		tmp = cos(y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(1.0, z, Float64(x * sin(y)))
	tmp = 0.0
	if (x <= -6e-61)
		tmp = t_0;
	elseif (x <= 2.7e-17)
		tmp = Float64(cos(y) * z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 * z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e-61], t$95$0, If[LessEqual[x, 2.7e-17], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000024e-61 or 2.7000000000000001e-17 < x

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.7%

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

   if -6.00000000000000024e-61 < x < 2.7000000000000001e-17

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 91.4

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

   5. Applied rewrites 91.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x \cdot \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -4.2e-128) t_0 (if (<= z 1.65e-65) (* x (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -4.2e-128) {
		tmp = t_0;
	} else if (z <= 1.65e-65) {
		tmp = x * sin(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 = cos(y) * z
    if (z <= (-4.2d-128)) then
        tmp = t_0
    else if (z <= 1.65d-65) then
        tmp = x * sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -4.2e-128) {
		tmp = t_0;
	} else if (z <= 1.65e-65) {
		tmp = x * Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -4.2e-128:
		tmp = t_0
	elif z <= 1.65e-65:
		tmp = x * math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -4.2e-128)
		tmp = t_0;
	elseif (z <= 1.65e-65)
		tmp = Float64(x * sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -4.2e-128)
		tmp = t_0;
	elseif (z <= 1.65e-65)
		tmp = x * sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.2e-128], t$95$0, If[LessEqual[z, 1.65e-65], N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e-128 or 1.6500000000000001e-65 < 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. lower-*.f64 N/A

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

      2. lower-cos.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -4.2000000000000002e-128 < z < 1.6500000000000001e-65

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-128}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -0.025:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.078:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot -0.5, x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))))
   (if (<= y -0.025)
     t_0
     (if (<= y 0.078) (fma y (fma z (* y -0.5) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double tmp;
	if (y <= -0.025) {
		tmp = t_0;
	} else if (y <= 0.078) {
		tmp = fma(y, fma(z, (y * -0.5), x), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	tmp = 0.0
	if (y <= -0.025)
		tmp = t_0;
	elseif (y <= 0.078)
		tmp = fma(y, fma(z, Float64(y * -0.5), x), z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.025], t$95$0, If[LessEqual[y, 0.078], N[(y * N[(z * N[(y * -0.5), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.025000000000000001 or 0.0779999999999999999 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.5

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

   5. Applied rewrites 50.5%

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

   if -0.025000000000000001 < y < 0.0779999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 41.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-163}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-66}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e-163) z (if (<= z 1.6e-66) (* y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-163) {
		tmp = z;
	} else if (z <= 1.6e-66) {
		tmp = y * x;
	} else {
		tmp = 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 (z <= (-8d-163)) then
        tmp = z
    else if (z <= 1.6d-66) then
        tmp = y * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e-163) {
		tmp = z;
	} else if (z <= 1.6e-66) {
		tmp = y * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e-163:
		tmp = z
	elif z <= 1.6e-66:
		tmp = y * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e-163)
		tmp = z;
	elseif (z <= 1.6e-66)
		tmp = Float64(y * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8e-163)
		tmp = z;
	elseif (z <= 1.6e-66)
		tmp = y * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e-163], z, If[LessEqual[z, 1.6e-66], N[(y * x), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999939e-163 or 1.59999999999999991e-66 < 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. lower-*.f64 N/A

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

      2. lower-cos.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto z \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 50.9

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

   10. Applied rewrites 50.9%

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

   if -7.99999999999999939e-163 < z < 1.59999999999999991e-66

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   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 45.7

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

   5. Applied rewrites 45.7%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-163}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-66}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% 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 53.4

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

5. Applied rewrites 53.4%

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

Alternative 7: 39.4% accurate, 214.0× 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 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. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in y around 0

   \[\leadsto z \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 38.7%

   \[\leadsto z \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 38.7

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

10. Applied rewrites 38.7%

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

Reproduce

herbie shell --seed 2024219 
(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))))