Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 89.6%

Time: 5.9s

Alternatives: 4

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

real(4) function code(alpha, u0)
    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: 56.0% 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

real(4) function code(alpha, u0)
    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: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{{\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\alpha \cdot \alpha\right) \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9998000264167786)
   (*
    (log (- 1.0 u0))
    (/
     (pow (* (- alpha) alpha) 3.0)
     (- (* (* alpha alpha) (* alpha alpha)) (* (* alpha alpha) 0.0))))
   (* (* alpha u0) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9998000264167786f) {
		tmp = logf((1.0f - u0)) * (powf((-alpha * alpha), 3.0f) / (((alpha * alpha) * (alpha * alpha)) - ((alpha * alpha) * 0.0f)));
	} else {
		tmp = (alpha * u0) * alpha;
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9998000264167786e0) then
        tmp = log((1.0e0 - u0)) * (((-alpha * alpha) ** 3.0e0) / (((alpha * alpha) * (alpha * alpha)) - ((alpha * alpha) * 0.0e0)))
    else
        tmp = (alpha * u0) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9998000264167786))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32((Float32(Float32(-alpha) * alpha) ^ Float32(3.0)) / Float32(Float32(Float32(alpha * alpha) * Float32(alpha * alpha)) - Float32(Float32(alpha * alpha) * Float32(0.0)))));
	else
		tmp = Float32(Float32(alpha * u0) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9998000264167786))
		tmp = log((single(1.0) - u0)) * (((-alpha * alpha) ^ single(3.0)) / (((alpha * alpha) * (alpha * alpha)) - ((alpha * alpha) * single(0.0))));
	else
		tmp = (alpha * u0) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.999800026

   1. Initial program 87.2%

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

      1. +-lft-identity N/A

         \[\leadsto \color{blue}{\left(0 + \left(-\alpha\right) \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 \cdot 0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)}} \cdot \log \left(1 - u0\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 \cdot 0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)}} \cdot \log \left(1 - u0\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 \cdot 0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)} \cdot \log \left(1 - u0\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \frac{\color{blue}{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}}{0 \cdot 0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)} \cdot \log \left(1 - u0\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \frac{0 + \color{blue}{{\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}}{0 \cdot 0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)} \cdot \log \left(1 - u0\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{\color{blue}{0} + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)} \cdot \log \left(1 - u0\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \frac{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{\color{blue}{0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)}} \cdot \log \left(1 - u0\right) \]

      9. lower--.f32 N/A

         \[\leadsto \frac{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 + \color{blue}{\left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)}} \cdot \log \left(1 - u0\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 + \left(\color{blue}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)} - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)} \cdot \log \left(1 - u0\right) \]

      11. lower-*.f32 87.2

          \[\leadsto \frac{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - \color{blue}{0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)}\right)} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\frac{0 + {\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{0 + \left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) - 0 \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\right)}} \cdot \log \left(1 - u0\right) \]

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 32.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 92.0

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

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.3%

      \[\leadsto \left(u0 \cdot \alpha\right) \cdot \color{blue}{\alpha} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998000264167786:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{{\left(\left(-\alpha\right) \cdot \alpha\right)}^{3}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\alpha \cdot \alpha\right) \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00019999999494757503:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00019999999494757503)
   (* (* alpha u0) alpha)
   (* (log (- 1.0 u0)) (* (- alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00019999999494757503f) {
		tmp = (alpha * u0) * alpha;
	} else {
		tmp = logf((1.0f - u0)) * (-alpha * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.00019999999494757503e0) then
        tmp = (alpha * u0) * alpha
    else
        tmp = log((1.0e0 - u0)) * (-alpha * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(alpha * u0) * alpha);
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00019999999494757503))
		tmp = (alpha * u0) * alpha;
	else
		tmp = log((single(1.0) - u0)) * (-alpha * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.99999995e-4

   1. Initial program 32.9%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 92.0

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

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.3%

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

   if 1.99999995e-4 < u0

   1. Initial program 87.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00019999999494757503:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 76.9

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

5. Applied rewrites 76.9%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
6. Step-by-step derivation

7. Applied rewrites 77.1%

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

8. Final simplification 77.1%

   \[\leadsto \left(\alpha \cdot u0\right) \cdot \alpha \]
9. Add Preprocessing

Alternative 4: 74.5% 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

real(4) function code(alpha, u0)
    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 52.0%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 76.9

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

5. Applied rewrites 76.9%

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

Reproduce

herbie shell --seed 2024288 
(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))))