Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 99.0%

Time: 4.4s

Alternatives: 5

Speedup: N/A×

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: 55.8% 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, N/A× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{log1p}\left(-1 \cdot u0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* -1.0 (* (* alpha alpha) (log1p (* -1.0 u0)))))

C

float code(float alpha, float u0) {
	return -1.0f * ((alpha * alpha) * log1pf((-1.0f * u0)));
}

Julia

function code(alpha, u0)
	return Float32(Float32(-1.0) * Float32(Float32(alpha * alpha) * log1p(Float32(Float32(-1.0) * u0))))
end

Derivation

1. Initial program 53.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. *-lft-identity N/A

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

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

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

   5. metadata-eval N/A

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. mul-1-neg N/A

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{-1 \cdot u0}\right) \]

   9. lower-*.f32 99.0

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{-1 \cdot u0}\right) \]

4. Applied rewrites 99.0%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \]

5. Final simplification 99.0%

   \[\leadsto -1 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \mathsf{log1p}\left(-1 \cdot u0\right)\right) \]
6. Add Preprocessing

Alternative 2: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;t\_0 \leq -0.041999999433755875:\\ \;\;\;\;-1 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \alpha, -1 \cdot \alpha, \left(\mathsf{fma}\left(t\_1 \cdot 0.25, u0, t\_1 \cdot 0.3333333333333333\right) - -0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))) (t_1 (* (* alpha alpha) u0)))
   (if (<= t_0 -0.041999999433755875)
     (* -1.0 (* (* alpha alpha) t_0))
     (*
      (fma
       (* -1.0 alpha)
       (* -1.0 alpha)
       (*
        (-
         (fma (* t_1 0.25) u0 (* t_1 0.3333333333333333))
         (* -0.5 (* alpha alpha)))
        u0))
      u0))))

C

float code(float alpha, float u0) {
	float t_0 = logf((1.0f - u0));
	float t_1 = (alpha * alpha) * u0;
	float tmp;
	if (t_0 <= -0.041999999433755875f) {
		tmp = -1.0f * ((alpha * alpha) * t_0);
	} else {
		tmp = fmaf((-1.0f * alpha), (-1.0f * alpha), ((fmaf((t_1 * 0.25f), u0, (t_1 * 0.3333333333333333f)) - (-0.5f * (alpha * alpha))) * u0)) * u0;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.041999999433755875))
		tmp = Float32(Float32(-1.0) * Float32(Float32(alpha * alpha) * t_0));
	else
		tmp = Float32(fma(Float32(Float32(-1.0) * alpha), Float32(Float32(-1.0) * alpha), Float32(Float32(fma(Float32(t_1 * Float32(0.25)), u0, Float32(t_1 * Float32(0.3333333333333333))) - Float32(Float32(-0.5) * Float32(alpha * alpha))) * u0)) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0419999994

   1. Initial program 97.2%

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

   if -0.0419999994 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 46.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(-1 \cdot \alpha, -1 \cdot \alpha, \left(\mathsf{fma}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.25, u0, \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.3333333333333333\right) - -0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \cdot u0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u0\right) \leq -0.041999999433755875:\\ \;\;\;\;-1 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \alpha, -1 \cdot \alpha, \left(\mathsf{fma}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.25, u0, \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.3333333333333333\right) - -0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \cdot u0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u0\right)\\ t_1 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathbf{if}\;t\_0 \leq -0.041999999433755875:\\ \;\;\;\;\left(t\_0 \cdot \left(-1 \cdot \alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \alpha, -1 \cdot \alpha, \left(\mathsf{fma}\left(t\_1 \cdot 0.25, u0, t\_1 \cdot 0.3333333333333333\right) - -0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \cdot u0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))) (t_1 (* (* alpha alpha) u0)))
   (if (<= t_0 -0.041999999433755875)
     (* (* t_0 (* -1.0 alpha)) alpha)
     (*
      (fma
       (* -1.0 alpha)
       (* -1.0 alpha)
       (*
        (-
         (fma (* t_1 0.25) u0 (* t_1 0.3333333333333333))
         (* -0.5 (* alpha alpha)))
        u0))
      u0))))

C

float code(float alpha, float u0) {
	float t_0 = logf((1.0f - u0));
	float t_1 = (alpha * alpha) * u0;
	float tmp;
	if (t_0 <= -0.041999999433755875f) {
		tmp = (t_0 * (-1.0f * alpha)) * alpha;
	} else {
		tmp = fmaf((-1.0f * alpha), (-1.0f * alpha), ((fmaf((t_1 * 0.25f), u0, (t_1 * 0.3333333333333333f)) - (-0.5f * (alpha * alpha))) * u0)) * u0;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	t_0 = log(Float32(Float32(1.0) - u0))
	t_1 = Float32(Float32(alpha * alpha) * u0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.041999999433755875))
		tmp = Float32(Float32(t_0 * Float32(Float32(-1.0) * alpha)) * alpha);
	else
		tmp = Float32(fma(Float32(Float32(-1.0) * alpha), Float32(Float32(-1.0) * alpha), Float32(Float32(fma(Float32(t_1 * Float32(0.25)), u0, Float32(t_1 * Float32(0.3333333333333333))) - Float32(Float32(-0.5) * Float32(alpha * alpha))) * u0)) * u0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u0)) < -0.0419999994

   1. Initial program 97.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

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

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

      5. unpow2 N/A

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

      6. mul-1-neg N/A

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

      7. lift--.f32 N/A

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

      8. lift-log.f32 N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

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

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

      13. associate-*r* N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-log.f32 N/A

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

      17. lift--.f32 N/A

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

      18. mul-1-neg N/A

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

      19. lower-*.f32 96.9

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

   4. Applied rewrites 96.9%

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

   if -0.0419999994 < (log.f32 (-.f32 #s(literal 1 binary32) u0))

   1. Initial program 46.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(-1 \cdot \alpha, -1 \cdot \alpha, \left(\mathsf{fma}\left(\left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.25, u0, \left(\left(\alpha \cdot \alpha\right) \cdot u0\right) \cdot 0.3333333333333333\right) - -0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \cdot u0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha \cdot \alpha\right) \cdot u0\\ \mathsf{fma}\left(-1 \cdot \alpha, -1 \cdot \alpha, \left(\mathsf{fma}\left(t\_0 \cdot 0.25, u0, t\_0 \cdot 0.3333333333333333\right) - -0.5 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot u0\right) \cdot u0 \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (* alpha alpha) u0)))
   (*
    (fma
     (* -1.0 alpha)
     (* -1.0 alpha)
     (*
      (-
       (fma (* t_0 0.25) u0 (* t_0 0.3333333333333333))
       (* -0.5 (* alpha alpha)))
      u0))
    u0)))

C

float code(float alpha, float u0) {
	float t_0 = (alpha * alpha) * u0;
	return fmaf((-1.0f * alpha), (-1.0f * alpha), ((fmaf((t_0 * 0.25f), u0, (t_0 * 0.3333333333333333f)) - (-0.5f * (alpha * alpha))) * u0)) * u0;
}

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(alpha * alpha) * u0)
	return Float32(fma(Float32(Float32(-1.0) * alpha), Float32(Float32(-1.0) * alpha), Float32(Float32(fma(Float32(t_0 * Float32(0.25)), u0, Float32(t_0 * Float32(0.3333333333333333))) - Float32(Float32(-0.5) * Float32(alpha * alpha))) * u0)) * u0)
end

Derivation

1. Initial program 53.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 92.9%

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

7. Applied rewrites 93.0%

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

Alternative 5: 92.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, 0.25, \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{u0}, 0.3333333333333333, \mathsf{fma}\left({\left(\frac{\alpha}{u0}\right)}^{2}, 0.5, \frac{\alpha \cdot \alpha}{\left(u0 \cdot u0\right) \cdot u0}\right)\right)\right) \cdot {\left(u0 \cdot u0\right)}^{2} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (fma
   (* alpha alpha)
   0.25
   (fma
    (/ (* alpha alpha) u0)
    0.3333333333333333
    (fma (pow (/ alpha u0) 2.0) 0.5 (/ (* alpha alpha) (* (* u0 u0) u0)))))
  (pow (* u0 u0) 2.0)))

C

float code(float alpha, float u0) {
	return fmaf((alpha * alpha), 0.25f, fmaf(((alpha * alpha) / u0), 0.3333333333333333f, fmaf(powf((alpha / u0), 2.0f), 0.5f, ((alpha * alpha) / ((u0 * u0) * u0))))) * powf((u0 * u0), 2.0f);
}

Julia

function code(alpha, u0)
	return Float32(fma(Float32(alpha * alpha), Float32(0.25), fma(Float32(Float32(alpha * alpha) / u0), Float32(0.3333333333333333), fma((Float32(alpha / u0) ^ Float32(2.0)), Float32(0.5), Float32(Float32(alpha * alpha) / Float32(Float32(u0 * u0) * u0))))) * (Float32(u0 * u0) ^ Float32(2.0)))
end

Derivation

1. Initial program 53.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 92.9%

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

6. Taylor expanded in u0 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 92.2%

   \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, 0.25, \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{u0}, 0.3333333333333333, \mathsf{fma}\left({\left(\frac{\alpha}{u0}\right)}^{2}, 0.5, \frac{\alpha \cdot \alpha}{\left(u0 \cdot u0\right) \cdot u0}\right)\right)\right) \cdot \color{blue}{{\left(u0 \cdot u0\right)}^{2}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))