Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 99.0%

Time: 9.8s

Alternatives: 11

Speedup: 21.6×

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: 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

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. distribute-lft-neg-out 54.5%

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

   3. distribute-rgt-neg-in 54.5%

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

   4. distribute-lft-neg-out 54.5%

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

   5. sub-neg 54.5%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 93.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (u0 * ((alpha * 0.5e0) + ((u0 * (u0 * (alpha * 0.25e0))) + (u0 * (alpha * 0.3333333333333333e0)))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (u0 * ((alpha * single(0.5)) + ((u0 * (u0 * (alpha * single(0.25)))) + (u0 * (alpha * single(0.3333333333333333))))))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 92.7%

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

   1. distribute-rgt-in 92.7%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \color{blue}{\left(\left(0.25 \cdot \left(\alpha \cdot u0\right)\right) \cdot u0 + \left(0.3333333333333333 \cdot \alpha\right) \cdot u0\right)}\right)\right)\right) \]

   2. associate-*r* 92.7%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \left(\color{blue}{\left(\left(0.25 \cdot \alpha\right) \cdot u0\right)} \cdot u0 + \left(0.3333333333333333 \cdot \alpha\right) \cdot u0\right)\right)\right)\right) \]

   3. *-commutative 92.7%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \left(\left(\left(0.25 \cdot \alpha\right) \cdot u0\right) \cdot u0 + \color{blue}{\left(\alpha \cdot 0.3333333333333333\right)} \cdot u0\right)\right)\right)\right) \]

7. Applied egg-rr 92.7%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \color{blue}{\left(\left(\left(0.25 \cdot \alpha\right) \cdot u0\right) \cdot u0 + \left(\alpha \cdot 0.3333333333333333\right) \cdot u0\right)}\right)\right)\right) \]

8. Final simplification 92.7%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5 + \left(u0 \cdot \left(u0 \cdot \left(\alpha \cdot 0.25\right)\right) + u0 \cdot \left(\alpha \cdot 0.3333333333333333\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 93.5% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (u0 * ((alpha * 0.5e0) + (u0 * ((0.25e0 * (alpha * u0)) + (alpha * 0.3333333333333333e0)))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (u0 * ((alpha * single(0.5)) + (u0 * ((single(0.25) * (alpha * u0)) + (alpha * single(0.3333333333333333))))))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 92.7%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Final simplification 92.7%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5 + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + \alpha \cdot 0.3333333333333333\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 93.5% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (u0 * ((alpha * 0.5e0) + (alpha * (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (u0 * ((alpha * single(0.5)) + (alpha * (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 92.7%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Taylor expanded in alpha around 0 92.7%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \color{blue}{\alpha \cdot \left(u0 \cdot \left(0.3333333333333333 + 0.25 \cdot u0\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 92.7%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \alpha \cdot \left(u0 \cdot \left(0.3333333333333333 + \color{blue}{u0 \cdot 0.25}\right)\right)\right)\right)\right) \]

8. Simplified 92.7%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + \color{blue}{\alpha \cdot \left(u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)}\right)\right)\right) \]

9. Final simplification 92.7%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5 + \alpha \cdot \left(u0 \cdot \left(0.3333333333333333 + u0 \cdot 0.25\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 93.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * (0.3333333333333333e0 + (u0 * 0.25e0))))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * (single(0.3333333333333333) + (u0 * single(0.25)))))))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 92.7%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.5 \cdot \alpha + u0 \cdot \left(0.25 \cdot \left(\alpha \cdot u0\right) + 0.3333333333333333 \cdot \alpha\right)\right)\right)\right)} \]

6. Taylor expanded in alpha around 0 92.5%

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

   1. *-commutative 92.5%

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

8. Simplified 92.5%

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

Alternative 6: 91.6% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (alpha * (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (alpha * (u0 * (single(0.5) + (u0 * single(0.3333333333333333)))))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 90.8%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.3333333333333333 \cdot \left(\alpha \cdot u0\right) + 0.5 \cdot \alpha\right)\right)\right)} \]

6. Taylor expanded in alpha around 0 90.8%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\alpha \cdot \left(u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 90.8%

      \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \alpha \cdot \left(u0 \cdot \left(0.5 + \color{blue}{u0 \cdot 0.3333333333333333}\right)\right)\right)\right) \]

8. Simplified 90.8%

   \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}\right)\right) \]
9. Add Preprocessing

Alternative 7: 91.3% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * (1.0e0 + (u0 * (0.5e0 + (u0 * 0.3333333333333333e0))))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) + (u0 * (single(0.5) + (u0 * single(0.3333333333333333)))))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 90.8%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(0.3333333333333333 \cdot \left(\alpha \cdot u0\right) + 0.5 \cdot \alpha\right)\right)\right)} \]

6. Taylor expanded in alpha around 0 90.6%

   \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right)\right)\right)} \]

7. Final simplification 90.6%

   \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 8: 87.5% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * ((alpha * u0) + (u0 * (u0 * (alpha * 0.5e0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * ((alpha * u0) + (u0 * (u0 * (alpha * single(0.5)))));
end

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. distribute-lft-neg-out 54.5%

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

   3. distribute-rgt-neg-in 54.5%

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

   4. distribute-lft-neg-out 54.5%

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

   5. sub-neg 54.5%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. neg-sub0 98.9%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{0 - u0}\right)\right) \]

   2. flip-- 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{0 \cdot 0 - u0 \cdot u0}{0 + u0}}\right)\right) \]

   3. metadata-eval 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{0} - u0 \cdot u0}{0 + u0}\right)\right) \]

   4. pow2 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - \color{blue}{{u0}^{2}}}{0 + u0}\right)\right) \]

   5. metadata-eval 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\color{blue}{\log 1} + u0}\right)\right) \]

   6. add-log-exp 49.6%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\log 1 + \color{blue}{\log \left(e^{u0}\right)}}\right)\right) \]

   7. log-prod 49.6%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\color{blue}{\log \left(1 \cdot e^{u0}\right)}}\right)\right) \]

   8. *-un-lft-identity 49.6%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\log \color{blue}{\left(e^{u0}\right)}}\right)\right) \]

   9. add-log-exp 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\color{blue}{u0}}\right)\right) \]

6. Applied egg-rr 98.7%

   \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{0 - {u0}^{2}}{u0}}\right)\right) \]
7. Step-by-step derivation

   1. sub0-neg 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{-{u0}^{2}}}{u0}\right)\right) \]

8. Simplified 98.7%

   \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-{u0}^{2}}{u0}}\right)\right) \]

9. Taylor expanded in u0 around 0 86.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\left(\alpha \cdot u0\right) \cdot 0.5}\right)\right) \]

    2. *-commutative 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\left(u0 \cdot \alpha\right)} \cdot 0.5\right)\right) \]

    3. associate-*r* 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{u0 \cdot \left(\alpha \cdot 0.5\right)}\right)\right) \]

11. Simplified 86.9%

    \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\alpha \cdot 0.5\right) + \alpha\right)}\right) \]

    2. distribute-rgt-in 87.0%

       \[\leadsto \alpha \cdot \color{blue}{\left(\left(u0 \cdot \left(\alpha \cdot 0.5\right)\right) \cdot u0 + \alpha \cdot u0\right)} \]

13. Applied egg-rr 87.0%

    \[\leadsto \alpha \cdot \color{blue}{\left(\left(u0 \cdot \left(\alpha \cdot 0.5\right)\right) \cdot u0 + \alpha \cdot u0\right)} \]

14. Final simplification 87.0%

    \[\leadsto \alpha \cdot \left(\alpha \cdot u0 + u0 \cdot \left(u0 \cdot \left(\alpha \cdot 0.5\right)\right)\right) \]
15. Add Preprocessing

Alternative 9: 87.5% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (u0 * (alpha + (0.5e0 * (alpha * u0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (u0 * (alpha + (single(0.5) * (alpha * u0))));
end

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 86.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right)} \]
6. Add Preprocessing

Alternative 10: 87.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * (u0 * (1.0e0 + (u0 * 0.5e0))))
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * (u0 * (single(1.0) + (u0 * single(0.5)))));
end

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. distribute-lft-neg-out 54.5%

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

   3. distribute-rgt-neg-in 54.5%

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

   4. distribute-lft-neg-out 54.5%

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

   5. sub-neg 54.5%

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

   6. log1p-define 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. neg-sub0 98.9%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{0 - u0}\right)\right) \]

   2. flip-- 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{0 \cdot 0 - u0 \cdot u0}{0 + u0}}\right)\right) \]

   3. metadata-eval 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{0} - u0 \cdot u0}{0 + u0}\right)\right) \]

   4. pow2 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - \color{blue}{{u0}^{2}}}{0 + u0}\right)\right) \]

   5. metadata-eval 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\color{blue}{\log 1} + u0}\right)\right) \]

   6. add-log-exp 49.6%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\log 1 + \color{blue}{\log \left(e^{u0}\right)}}\right)\right) \]

   7. log-prod 49.6%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\color{blue}{\log \left(1 \cdot e^{u0}\right)}}\right)\right) \]

   8. *-un-lft-identity 49.6%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\log \color{blue}{\left(e^{u0}\right)}}\right)\right) \]

   9. add-log-exp 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{0 - {u0}^{2}}{\color{blue}{u0}}\right)\right) \]

6. Applied egg-rr 98.7%

   \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{0 - {u0}^{2}}{u0}}\right)\right) \]
7. Step-by-step derivation

   1. sub0-neg 98.7%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{-{u0}^{2}}}{u0}\right)\right) \]

8. Simplified 98.7%

   \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-{u0}^{2}}{u0}}\right)\right) \]

9. Taylor expanded in u0 around 0 86.9%

   \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\left(\alpha \cdot u0\right) \cdot 0.5}\right)\right) \]

    2. *-commutative 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{\left(u0 \cdot \alpha\right)} \cdot 0.5\right)\right) \]

    3. associate-*r* 86.9%

       \[\leadsto \alpha \cdot \left(u0 \cdot \left(\alpha + \color{blue}{u0 \cdot \left(\alpha \cdot 0.5\right)}\right)\right) \]

11. Simplified 86.9%

    \[\leadsto \alpha \cdot \color{blue}{\left(u0 \cdot \left(\alpha + u0 \cdot \left(\alpha \cdot 0.5\right)\right)\right)} \]

12. Taylor expanded in alpha around 0 86.8%

    \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(1 + 0.5 \cdot u0\right)\right)\right)} \]
13. Step-by-step derivation

    1. *-commutative 86.8%

       \[\leadsto \alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + \color{blue}{u0 \cdot 0.5}\right)\right)\right) \]

14. Simplified 86.8%

    \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)\right)} \]
15. Add Preprocessing

Alternative 11: 74.5% accurate, 21.6× speedup

Math

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

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(alpha * Float32(alpha * u0))
end

MATLAB

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

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in u0 around 0 74.8%

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

Reproduce

herbie shell --seed 2024145 
(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))))