Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Percentage Accurate: 60.8% → 98.4%

Time: 14.3s

Alternatives: 14

Speedup: 3.2×

Specification

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

\[\begin{array}{l} \\ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Initial Program: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (- cos2phi) (* alphax alphax)) (/ (/ sin2phi alphay) alphay))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((-cos2phi / (alphax * alphax)) - ((sin2phi / alphay) / alphay));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(Float32(sin2phi / alphay) / alphay)))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-log1p.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower-neg.f32 98.7

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{-u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 98.7%

   \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

   3. lower-/.f32 98.8

      \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay}} \]

6. Applied rewrites 98.8%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}} \]

7. Final simplification 98.8%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{\frac{sin2phi}{alphay}}{alphay}} \]
8. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(-u0\right)}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (log1p (- u0))
  (- (/ (- cos2phi) (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return log1pf(-u0) / ((-cos2phi / (alphax * alphax)) - (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(log1p(Float32(-u0)) / Float32(Float32(Float32(-cos2phi) / Float32(alphax * alphax)) - Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-log1p.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower-neg.f32 98.7

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{-u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 98.7%

   \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Final simplification 98.7%

   \[\leadsto \frac{\mathsf{log1p}\left(-u0\right)}{\frac{-cos2phi}{alphax \cdot alphax} - \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 3: 93.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (fma u0 (* u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5)) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf(u0, (u0 * fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f)), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(u0, Float32(u0 * fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5))), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 91.2

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 91.2%

   \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + u0 \cdot 1}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right) + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{3} + \frac{1}{4} \cdot u0, \frac{1}{2}\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, \frac{1}{2}\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   10. lower-fma.f32 93.3

       \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Applied rewrites 93.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 4: 92.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (fma u0 (fma u0 (fma u0 0.25 0.3333333333333333) 0.5) 1.0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * fmaf(u0, fmaf(u0, fmaf(u0, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * fma(u0, fma(u0, fma(u0, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 87.4

      \[\leadsto \frac{-u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 87.4%

   \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right)\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} + u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u0\right) + \frac{1}{2}}, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{1}{3} + \frac{1}{4} \cdot u0, \frac{1}{2}\right)}, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{1}{4} \cdot u0 + \frac{1}{3}}, \frac{1}{2}\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 93.2

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Applied rewrites 93.2%

   \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 5: 91.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (fma u0 (* u0 (fma u0 0.3333333333333333 0.5)) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf(u0, (u0 * fmaf(u0, 0.3333333333333333f, 0.5f)), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(u0, Float32(u0 * fma(u0, Float32(0.3333333333333333), Float32(0.5))), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 91.2

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 91.2%

   \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) + u0 \cdot 1}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{u0 \cdot \left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right) + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \color{blue}{\left(\frac{1}{3} \cdot u0 + \frac{1}{2}\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \left(\color{blue}{u0 \cdot \frac{1}{3}} + \frac{1}{2}\right), u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 91.3

      \[\leadsto \frac{\mathsf{fma}\left(u0, u0 \cdot \color{blue}{\mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Applied rewrites 91.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, u0 \cdot \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 6: 87.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (fma u0 (* u0 0.5) u0)
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf(u0, (u0 * 0.5f), u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(\frac{1}{2} \cdot u0\right) + u0 \cdot 1}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{u0 \cdot \left(\frac{1}{2} \cdot u0\right) + \color{blue}{u0}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} \cdot u0, u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{2}}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. lower-*.f32 87.5

      \[\leadsto \frac{\mathsf{fma}\left(u0, \color{blue}{u0 \cdot 0.5}, u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 87.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
6. Add Preprocessing

Alternative 7: 87.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (* u0 (fma u0 0.5 1.0))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * fmaf(u0, 0.5f, 1.0f)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. lower-fma.f32 87.4

      \[\leadsto \frac{-u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 87.4%

   \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. *-commutative N/A

      \[\leadsto \frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 87.4

      \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

8. Applied rewrites 87.4%

   \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
9. Add Preprocessing

Alternative 8: 87.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (*
  (fma u0 0.5 1.0)
  (/ u0 (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf(u0, 0.5f, 1.0f) * (u0 / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay))));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(u0, Float32(0.5), Float32(1.0)) * Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay)))))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{u0 \cdot \left(\frac{1}{2} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + u0 \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(u0 \cdot \frac{1}{2}\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} + u0 \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot u0\right)} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + u0 \cdot \frac{1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   4. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{2} \cdot u0\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \color{blue}{\frac{u0 \cdot 1}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{1}{2} \cdot u0\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{\color{blue}{u0}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   6. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot u0\right)} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   8. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot u0\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   9. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right) \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   11. lower-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2}, 1\right)} \cdot \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} \]

   12. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \frac{1}{2}, 1\right) \cdot \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   13. lower-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \frac{1}{2}, 1\right) \cdot \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

   14. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \frac{1}{2}, 1\right) \cdot \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(u0, \frac{1}{2}, 1\right) \cdot \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   16. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \frac{1}{2}, 1\right) \cdot \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

   17. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(u0, \frac{1}{2}, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

5. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right) \cdot \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]
6. Add Preprocessing

Alternative 9: 82.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 0.004000000189989805:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 0.004000000189989805)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (- (/ (* u0 (fma u0 (fma u0 -0.3333333333333333 -0.5) -1.0)) t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (sin2phi <= 0.004000000189989805f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = -((u0 * fmaf(u0, fmaf(u0, -0.3333333333333333f, -0.5f), -1.0f)) / t_0);
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.004000000189989805))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(-Float32(Float32(u0 * fma(u0, fma(u0, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))) / t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.00400000019

   1. Initial program 54.5%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. lower-*.f32 75.6

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   if 0.00400000019 < sin2phi

   1. Initial program 65.3%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-fma.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 91.4

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 91.4%

      \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      3. lower-*.f32 90.8

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   8. Applied rewrites 90.8%

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.004000000189989805:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;sin2phi \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (/ sin2phi (* alphay alphay))))
   (if (<= sin2phi 0.019999999552965164)
     (/ u0 (+ (/ cos2phi (* alphax alphax)) t_0))
     (/ (* u0 (fma u0 -0.5 -1.0)) (- t_0)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = sin2phi / (alphay * alphay);
	float tmp;
	if (sin2phi <= 0.019999999552965164f) {
		tmp = u0 / ((cos2phi / (alphax * alphax)) + t_0);
	} else {
		tmp = (u0 * fmaf(u0, -0.5f, -1.0f)) / -t_0;
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(sin2phi / Float32(alphay * alphay))
	tmp = Float32(0.0)
	if (sin2phi <= Float32(0.019999999552965164))
		tmp = Float32(u0 / Float32(Float32(cos2phi / Float32(alphax * alphax)) + t_0));
	else
		tmp = Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(-t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 0.0199999996

   1. Initial program 55.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      2. lower-+.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}} + \frac{sin2phi}{{alphay}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{{alphay}^{2}}} \]

      6. lower-/.f32 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      8. lower-*.f32 75.2

         \[\leadsto \frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   if 0.0199999996 < sin2phi

   1. Initial program 65.0%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 87.8

         \[\leadsto \frac{-u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 87.8%

      \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      3. lower-*.f32 87.8

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   8. Applied rewrites 87.8%

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 0.019999999552965164:\\ \;\;\;\;\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.99999987306209e-21)
   (/
    (* u0 (fma u0 (fma u0 0.3333333333333333 0.5) 1.0))
    (/ cos2phi (* alphax alphax)))
   (/ (* u0 (fma u0 -0.5 -1.0)) (- (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.99999987306209e-21f) {
		tmp = (u0 * fmaf(u0, fmaf(u0, 0.3333333333333333f, 0.5f), 1.0f)) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (u0 * fmaf(u0, -0.5f, -1.0f)) / -(sin2phi / (alphay * alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.99999987306209e-21))
		tmp = Float32(Float32(u0 * fma(u0, fma(u0, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0))) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(-Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.9999999e-21

   1. Initial program 60.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-fma.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 88.6

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 88.6%

      \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      3. lower-*.f32 70.5

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   8. Applied rewrites 70.5%

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

       2. +-commutative N/A

          \[\leadsto \frac{u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u0\right) + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

       3. lower-fma.f32 N/A

          \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{1}{2} + \frac{1}{3} \cdot u0, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

       4. +-commutative N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{1}{3} \cdot u0 + \frac{1}{2}}, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

       5. *-commutative N/A

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{1}{3}} + \frac{1}{2}, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

       6. lower-fma.f32 70.5

          \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right)}, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

   11. Applied rewrites 70.5%

       \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

   if 3.9999999e-21 < sin2phi

   1. Initial program 60.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 88.7

         \[\leadsto \frac{-u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 88.7%

      \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      3. lower-*.f32 80.6

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   8. Applied rewrites 80.6%

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, 0.3333333333333333, 0.5\right), 1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= sin2phi 3.99999987306209e-21)
   (/ (* u0 (fma u0 0.5 1.0)) (/ cos2phi (* alphax alphax)))
   (/ (* u0 (fma u0 -0.5 -1.0)) (- (/ sin2phi (* alphay alphay))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if (sin2phi <= 3.99999987306209e-21f) {
		tmp = (u0 * fmaf(u0, 0.5f, 1.0f)) / (cos2phi / (alphax * alphax));
	} else {
		tmp = (u0 * fmaf(u0, -0.5f, -1.0f)) / -(sin2phi / (alphay * alphay));
	}
	return tmp;
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = Float32(0.0)
	if (sin2phi <= Float32(3.99999987306209e-21))
		tmp = Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(cos2phi / Float32(alphax * alphax)));
	else
		tmp = Float32(Float32(u0 * fma(u0, Float32(-0.5), Float32(-1.0))) / Float32(-Float32(sin2phi / Float32(alphay * alphay))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if sin2phi < 3.9999999e-21

   1. Initial program 60.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. lower-fma.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      8. lower-fma.f32 88.6

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 88.6%

      \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around inf

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

      3. lower-*.f32 70.5

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   8. Applied rewrites 70.5%

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   9. Taylor expanded in u0 around 0

      \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

       2. +-commutative N/A

          \[\leadsto \frac{u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

       3. *-commutative N/A

          \[\leadsto \frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

       4. lower-fma.f32 66.9

          \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

   11. Applied rewrites 66.9%

       \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

   if 3.9999999e-21 < sin2phi

   1. Initial program 60.2%

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(\frac{-1}{2} \cdot u0 - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u0 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(\color{blue}{u0 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

      5. lower-fma.f32 88.7

         \[\leadsto \frac{-u0 \cdot \color{blue}{\mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. Applied rewrites 88.7%

      \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. Taylor expanded in cos2phi around 0

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\color{blue}{\frac{sin2phi}{{alphay}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{2}, -1\right)\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

      3. lower-*.f32 80.6

         \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   8. Applied rewrites 80.6%

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sin2phi \leq 3.99999987306209 \cdot 10^{-21}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 \cdot \mathsf{fma}\left(u0, -0.5, -1\right)}{-\frac{sin2phi}{alphay \cdot alphay}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 25.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{\frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ (fma u0 (* u0 0.5) u0) (/ cos2phi (* alphax alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return fmaf(u0, (u0 * 0.5f), u0) / (cos2phi / (alphax * alphax));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(fma(u0, Float32(u0 * Float32(0.5)), u0) / Float32(cos2phi / Float32(alphax * alphax)))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u0\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lower-log1p.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lower-neg.f32 98.7

      \[\leadsto \frac{-\mathsf{log1p}\left(\color{blue}{-u0}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

4. Applied rewrites 98.7%

   \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(-u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{\color{blue}{alphay \cdot alphay}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{sin2phi}{alphay \cdot alphay}}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay} + \frac{cos2phi}{alphax \cdot alphax}}} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\frac{sin2phi}{\color{blue}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   9. div-inv N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\color{blue}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   10. lower-fma.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, \frac{1}{alphay}, \frac{cos2phi}{alphax \cdot alphax}\right)}} \]

   11. lower-/.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u0\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{\frac{sin2phi}{alphay}}, \frac{1}{alphay}, \frac{cos2phi}{alphax \cdot alphax}\right)} \]

   12. lower-/.f32 98.6

       \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, \color{blue}{\frac{1}{alphay}}, \frac{cos2phi}{alphax \cdot alphax}\right)} \]

6. Applied rewrites 98.6%

   \[\leadsto \frac{-\mathsf{log1p}\left(-u0\right)}{\color{blue}{\mathsf{fma}\left(\frac{sin2phi}{alphay}, \frac{1}{alphay}, \frac{cos2phi}{alphax \cdot alphax}\right)}} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\log \color{blue}{\left(1 - u0\right)}\right)}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   2. lift-approx N/A

      \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\color{blue}{\frac{sin2phi}{alphay}} \cdot \frac{1}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\frac{sin2phi}{alphay} \cdot \color{blue}{\frac{1}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} + \frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\frac{sin2phi}{alphay} \cdot \frac{1}{alphay} + \color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\frac{sin2phi}{alphay} \cdot \color{blue}{\frac{1}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   8. un-div-inv N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\frac{\color{blue}{\frac{sin2phi}{alphay}}}{alphay} + \frac{cos2phi}{alphax \cdot alphax}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\color{blue}{\frac{sin2phi}{alphay \cdot alphay}} + \frac{cos2phi}{alphax \cdot alphax}} \]

   11. +-commutative N/A

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}} \]

   12. lift-/.f32 N/A

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}} + \frac{sin2phi}{alphay \cdot alphay}} \]

   14. lift-approx N/A

       \[\leadsto \frac{u0 \cdot \mathsf{fma}\left(u0, \frac{1}{2}, 1\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

   15. lift-/.f32 27.0

       \[\leadsto \color{blue}{\frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}}} \]

8. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u0, u0 \cdot 0.5, u0\right)}{\frac{cos2phi}{alphax \cdot alphax}}} \]
9. Add Preprocessing

Alternative 14: 25.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}{\frac{cos2phi}{alphax \cdot alphax}} \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/ (* u0 (fma u0 0.5 1.0)) (/ cos2phi (* alphax alphax))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return (u0 * fmaf(u0, 0.5f, 1.0f)) / (cos2phi / (alphax * alphax));
}

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(u0 * fma(u0, Float32(0.5), Float32(1.0))) / Float32(cos2phi / Float32(alphax * alphax)))
end

Derivation

1. Initial program 60.2%

   \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) - 1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{3} \cdot u0 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{3} \cdot u0 - \frac{1}{2}, -1\right)}\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{3} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   8. lower-fma.f32 91.2

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right)}, -1\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

5. Applied rewrites 91.2%

   \[\leadsto \frac{-\color{blue}{u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

6. Taylor expanded in cos2phi around inf

   \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\color{blue}{\frac{cos2phi}{{alphax}^{2}}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\mathsf{neg}\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

   3. lower-*.f32 27.9

      \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\frac{cos2phi}{\color{blue}{alphax \cdot alphax}}} \]

8. Applied rewrites 27.9%

   \[\leadsto \frac{-u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.3333333333333333, -0.5\right), -1\right)}{\color{blue}{\frac{cos2phi}{alphax \cdot alphax}}} \]

9. Taylor expanded in u0 around 0

   \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{u0 \cdot \left(1 + \frac{1}{2} \cdot u0\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

    2. +-commutative N/A

       \[\leadsto \frac{u0 \cdot \color{blue}{\left(\frac{1}{2} \cdot u0 + 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

    3. *-commutative N/A

       \[\leadsto \frac{u0 \cdot \left(\color{blue}{u0 \cdot \frac{1}{2}} + 1\right)}{\frac{cos2phi}{alphax \cdot alphax}} \]

    4. lower-fma.f32 27.0

       \[\leadsto \frac{u0 \cdot \color{blue}{\mathsf{fma}\left(u0, 0.5, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]

11. Applied rewrites 27.0%

    \[\leadsto \frac{\color{blue}{u0 \cdot \mathsf{fma}\left(u0, 0.5, 1\right)}}{\frac{cos2phi}{alphax \cdot alphax}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))