Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 99.0%

Time: 7.7s

Alternatives: 11

Speedup: 10.5×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

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(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Initial Program: 56.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

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(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* (- alpha) (log1p (- u0))) alpha))

C

float code(float alpha, float u0) {
	return (-alpha * log1pf(-u0)) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * log1p(Float32(-u0))) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]
6. Add Preprocessing

Alternative 2: 93.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5 \cdot \alpha\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (*
   (fma
    (fma (* alpha (fma 0.25 u0 0.3333333333333333)) u0 (* 0.5 alpha))
    u0
    alpha)
   u0)
  alpha))

C

float code(float alpha, float u0) {
	return (fmaf(fmaf((alpha * fmaf(0.25f, u0, 0.3333333333333333f)), u0, (0.5f * alpha)), u0, alpha) * u0) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(fma(fma(Float32(alpha * fma(Float32(0.25), u0, Float32(0.3333333333333333))), u0, Float32(Float32(0.5) * alpha)), u0, alpha) * u0) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{2} \cdot \alpha + u0 \cdot \left(\frac{1}{4} \cdot \left(\alpha \cdot u0\right) + \frac{1}{3} \cdot \alpha\right)\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 93.2%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), u0, 0.5 \cdot \alpha\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]
9. Add Preprocessing

Alternative 3: 93.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (fma (* (fma 0.25 u0 0.3333333333333333) u0) u0 (fma 0.5 u0 1.0))
  (* (* alpha alpha) u0)))

C

float code(float alpha, float u0) {
	return fmaf((fmaf(0.25f, u0, 0.3333333333333333f) * u0), u0, fmaf(0.5f, u0, 1.0f)) * ((alpha * alpha) * u0);
}

Julia

function code(alpha, u0)
	return Float32(fma(Float32(fma(Float32(0.25), u0, Float32(0.3333333333333333)) * u0), u0, fma(Float32(0.5), u0, Float32(1.0))) * Float32(Float32(alpha * alpha) * u0))
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right) \cdot u0} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right) \cdot u0} \]

5. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u0 \cdot u0, \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), \mathsf{fma}\left(0.5, u0, 1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0} \]
6. Step-by-step derivation

7. Applied rewrites 92.9%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot \alpha\right) \cdot \alpha, u0 \cdot u0, \left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot \alpha\right) \cdot \alpha\right) \cdot u0 \]

8. Taylor expanded in alpha around 0

   \[\leadsto {\alpha}^{2} \cdot \color{blue}{\left(u0 \cdot \left(1 + \left(\frac{1}{2} \cdot u0 + {u0}^{2} \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 92.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right) \cdot u0, u0, \mathsf{fma}\left(0.5, u0, 1\right)\right) \cdot \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot u0\right)} \]
11. Add Preprocessing

Alternative 4: 93.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0 \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (*
   (fma u0 (fma u0 (fma 0.25 u0 0.3333333333333333) 0.5) 1.0)
   (* alpha alpha))
  u0))

C

float code(float alpha, float u0) {
	return (fmaf(u0, fmaf(u0, fmaf(0.25f, u0, 0.3333333333333333f), 0.5f), 1.0f) * (alpha * alpha)) * u0;
}

Julia

function code(alpha, u0)
	return Float32(Float32(fma(u0, fma(u0, fma(Float32(0.25), u0, Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)) * Float32(alpha * alpha)) * u0)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} \cdot {\alpha}^{2} + u0 \cdot \left(\frac{1}{4} \cdot \left({\alpha}^{2} \cdot u0\right) + \frac{1}{3} \cdot {\alpha}^{2}\right)\right) + {\alpha}^{2}\right)} \]

8. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(0.25, u0, 0.3333333333333333\right), 0.5\right), 1\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0} \]
9. Add Preprocessing

Alternative 5: 91.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u0, \alpha, \left(\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) \cdot \alpha\right) \cdot u0\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (fma u0 alpha (* (* (* (fma 0.3333333333333333 u0 0.5) alpha) u0) u0))
  alpha))

C

float code(float alpha, float u0) {
	return fmaf(u0, alpha, (((fmaf(0.3333333333333333f, u0, 0.5f) * alpha) * u0) * u0)) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(fma(u0, alpha, Float32(Float32(Float32(fma(Float32(0.3333333333333333), u0, Float32(0.5)) * alpha) * u0) * u0)) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 91.4%

   \[\leadsto \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]
9. Step-by-step derivation

10. Applied rewrites 91.5%

    \[\leadsto \mathsf{fma}\left(u0, \alpha, \left(\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) \cdot \alpha\right) \cdot u0\right) \cdot u0\right) \cdot \alpha \]
11. Add Preprocessing

Alternative 6: 91.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (fma (* alpha (fma 0.3333333333333333 u0 0.5)) u0 alpha) u0) alpha))

C

float code(float alpha, float u0) {
	return (fmaf((alpha * fmaf(0.3333333333333333f, u0, 0.5f)), u0, alpha) * u0) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(fma(Float32(alpha * fma(Float32(0.3333333333333333), u0, Float32(0.5))), u0, alpha) * u0) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 91.4%

   \[\leadsto \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]
9. Add Preprocessing

Alternative 7: 91.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (* (fma (fma 0.3333333333333333 u0 0.5) u0 1.0) alpha) u0) alpha))

C

float code(float alpha, float u0) {
	return ((fmaf(fmaf(0.3333333333333333f, u0, 0.5f), u0, 1.0f) * alpha) * u0) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(fma(fma(Float32(0.3333333333333333), u0, Float32(0.5)), u0, Float32(1.0)) * alpha) * u0) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 91.4%

   \[\leadsto \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]

9. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
10. Step-by-step derivation

11. Applied rewrites 87.4%

    \[\leadsto \left(\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \]

12. Taylor expanded in u0 around 0

    \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
13. Step-by-step derivation

14. Applied rewrites 91.3%

    \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \]
15. Add Preprocessing

Alternative 8: 87.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u0, \alpha, \left(\left(0.5 \cdot u0\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (fma u0 alpha (* (* (* 0.5 u0) alpha) u0)) alpha))

C

float code(float alpha, float u0) {
	return fmaf(u0, alpha, (((0.5f * u0) * alpha) * u0)) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(fma(u0, alpha, Float32(Float32(Float32(Float32(0.5) * u0) * alpha) * u0)) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 91.4%

   \[\leadsto \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]

9. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
10. Step-by-step derivation

11. Applied rewrites 87.4%

    \[\leadsto \left(\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \]
12. Step-by-step derivation

13. Applied rewrites 87.6%

    \[\leadsto \mathsf{fma}\left(u0, \alpha, \left(\left(0.5 \cdot u0\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \]
14. Add Preprocessing

Alternative 9: 87.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.5 \cdot \alpha, u0, \alpha\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (fma (* 0.5 alpha) u0 alpha) u0) alpha))

C

float code(float alpha, float u0) {
	return (fmaf((0.5f * alpha), u0, alpha) * u0) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(fma(Float32(Float32(0.5) * alpha), u0, alpha) * u0) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 91.4%

   \[\leadsto \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]

9. Taylor expanded in u0 around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2} \cdot \alpha, u0, \alpha\right) \cdot u0\right) \cdot \alpha \]
10. Step-by-step derivation

11. Applied rewrites 87.5%

    \[\leadsto \left(\mathsf{fma}\left(0.5 \cdot \alpha, u0, \alpha\right) \cdot u0\right) \cdot \alpha \]
12. Add Preprocessing

Alternative 10: 87.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (* (fma 0.5 u0 1.0) alpha) u0) alpha))

C

float code(float alpha, float u0) {
	return ((fmaf(0.5f, u0, 1.0f) * alpha) * u0) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(fma(Float32(0.5), u0, Float32(1.0)) * alpha) * u0) * alpha)
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right) \cdot \alpha}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\alpha \cdot \log \left(1 - u0\right)\right)\right) \cdot \alpha} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right) \cdot \alpha} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + u0 \cdot \left(\frac{1}{3} \cdot \left(\alpha \cdot u0\right) + \frac{1}{2} \cdot \alpha\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 91.4%

   \[\leadsto \left(\mathsf{fma}\left(\alpha \cdot \mathsf{fma}\left(0.3333333333333333, u0, 0.5\right), u0, \alpha\right) \cdot u0\right) \cdot \alpha \]

9. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
10. Step-by-step derivation

11. Applied rewrites 87.4%

    \[\leadsto \left(\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \]
12. Add Preprocessing

Alternative 11: 74.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot u0 \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* alpha alpha) u0))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * u0;
}

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(4) function code(alpha, u0)
use fmin_fmax_functions
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (alpha * alpha) * u0
end function

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * u0)
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (alpha * alpha) * u0;
end

Derivation

1. Initial program 56.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

   3. lower-*.f32 74.4

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot u0 \]

5. Applied rewrites 74.4%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))