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

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. remove-double-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. lift-sin.f64 N/A

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

   8. sin-neg-rev N/A

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

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

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

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

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. *-commutative N/A

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

   13. fp-cancel-sub-sign-inv N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. sin-neg-rev N/A

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

   17. lift-sin.f64 N/A

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

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

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

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

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

   20. lift-*.f64 N/A

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

   21. remove-double-neg N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 82.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \cos y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e-68)
   (fma (sin y) z (* 1.0 x))
   (if (<= z 1.9e-51) (fma y z (* (cos y) x)) (+ (* x 1.0) (* z (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e-68) {
		tmp = fma(sin(y), z, (1.0 * x));
	} else if (z <= 1.9e-51) {
		tmp = fma(y, z, (cos(y) * x));
	} else {
		tmp = (x * 1.0) + (z * sin(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e-68)
		tmp = fma(sin(y), z, Float64(1.0 * x));
	elseif (z <= 1.9e-51)
		tmp = fma(y, z, Float64(cos(y) * x));
	else
		tmp = Float64(Float64(x * 1.0) + Float64(z * sin(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e-68], N[(N[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e-51], N[(y * z + N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.40000000000000018e-68

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.7%

      \[\leadsto x \cdot \color{blue}{1} + z \cdot \sin y \]
   5. 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 76.7

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

      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 76.7

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

   6. Applied rewrites 76.7%

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

   if -3.40000000000000018e-68 < z < 1.90000000000000001e-51

   1. Initial program 99.8%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 65.0%

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

   if 1.90000000000000001e-51 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.7%

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

Alternative 3: 82.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, z, 1 \cdot x\right)\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \cos y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) z (* 1.0 x))))
   (if (<= z -3.4e-68) t_0 (if (<= z 1.9e-51) (fma y z (* (cos y) x)) 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 <= -3.4e-68) {
		tmp = t_0;
	} else if (z <= 1.9e-51) {
		tmp = fma(y, z, (cos(y) * x));
	} 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 <= -3.4e-68)
		tmp = t_0;
	elseif (z <= 1.9e-51)
		tmp = fma(y, z, Float64(cos(y) * x));
	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, -3.4e-68], t$95$0, If[LessEqual[z, 1.9e-51], N[(y * z + N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000018e-68 or 1.90000000000000001e-51 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.7%

      \[\leadsto x \cdot \color{blue}{1} + z \cdot \sin y \]
   5. 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 76.7

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

      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 76.7

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

   6. Applied rewrites 76.7%

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

   if -3.40000000000000018e-68 < z < 1.90000000000000001e-51

   1. Initial program 99.8%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 65.0%

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

Alternative 4: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin y, z, 1 \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (sin y) z (* 1.0 x)))

C

double code(double x, double y, double z) {
	return fma(sin(y), z, (1.0 * x));
}

Julia

function code(x, y, z)
	return fma(sin(y), z, Float64(1.0 * x))
end

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] * z + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 76.7%

   \[\leadsto x \cdot \color{blue}{1} + z \cdot \sin y \]
5. 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 76.7

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

   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 76.7

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

6. Applied rewrites 76.7%

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

Alternative 5: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;y \leq -0.19:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \mathsf{fma}\left(-0.5, x, -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= y -0.19)
     t_0
     (if (<= y 1.85e-7)
       (+ x (* y (+ z (* y (fma -0.5 x (* -0.16666666666666666 (* y z)))))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (y <= -0.19) {
		tmp = t_0;
	} else if (y <= 1.85e-7) {
		tmp = x + (y * (z + (y * fma(-0.5, x, (-0.16666666666666666 * (y * z))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (y <= -0.19)
		tmp = t_0;
	elseif (y <= 1.85e-7)
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * fma(-0.5, x, Float64(-0.16666666666666666 * Float64(y * z)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.19], t$95$0, If[LessEqual[y, 1.85e-7], N[(x + N[(y * N[(z + N[(y * N[(-0.5 * x + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.19 or 1.85000000000000002e-7 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 40.6

         \[\leadsto z \cdot \sin y \]

   4. Applied rewrites 40.6%

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

   if -0.19 < y < 1.85000000000000002e-7

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 51.2

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

   4. Applied rewrites 51.2%

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

Alternative 6: 52.3% accurate, 12.3× 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. Taylor expanded in y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 52.3

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

4. Applied rewrites 52.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

      \[\leadsto y \cdot z + x \]

   4. *-commutative N/A

      \[\leadsto z \cdot y + x \]

   5. lower-fma.f64 52.3

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

6. Applied rewrites 52.3%

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

Alternative 7: 17.0% accurate, 18.5× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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. Taylor expanded in y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 52.3

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 17.0

      \[\leadsto y \cdot z \]

7. Applied rewrites 17.0%

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

Reproduce

herbie shell --seed 2025151 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))