GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 9.2s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t\_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((t_0 * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	return Float32(t_0 / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(t_0 * cosTheta) * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) - single(1.0);
	tmp = t_0 / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t\_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((t_0 * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	return Float32(t_0 / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(t_0 * cosTheta) * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) - single(1.0);
	tmp = t_0 / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha + -1}{2 \cdot \left(\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) + \left(cosTheta \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \alpha \cdot \pi\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (+ (* alpha alpha) -1.0)
  (*
   2.0
   (+
    (* (log alpha) (* PI (- 1.0 (* cosTheta cosTheta))))
    (* (* cosTheta cosTheta) (* (* alpha alpha) (* (log alpha) PI)))))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / (2.0f * ((logf(alpha) * (((float) M_PI) * (1.0f - (cosTheta * cosTheta)))) + ((cosTheta * cosTheta) * ((alpha * alpha) * (logf(alpha) * ((float) M_PI))))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(2.0) * Float32(Float32(log(alpha) * Float32(Float32(pi) * Float32(Float32(1.0) - Float32(cosTheta * cosTheta)))) + Float32(Float32(cosTheta * cosTheta) * Float32(Float32(alpha * alpha) * Float32(log(alpha) * Float32(pi)))))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / (single(2.0) * ((log(alpha) * (single(pi) * (single(1.0) - (cosTheta * cosTheta)))) + ((cosTheta * cosTheta) * ((alpha * alpha) * (log(alpha) * single(pi))))));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) + 2 \cdot \left({\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) + {\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) + {\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right), \color{blue}{\left({\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \left(\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) + \left(cosTheta \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\pi \cdot \log \alpha\right)\right)\right)}} \]

6. Final simplification 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{2 \cdot \left(\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) + \left(cosTheta \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \alpha \cdot \pi\right)\right)\right)} \]
7. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t\_0}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + cosTheta \cdot \left(t\_0 \cdot cosTheta\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* cosTheta (* t_0 cosTheta)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return t_0 / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + (cosTheta * (t_0 * cosTheta))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) + Float32(-1.0))
	return Float32(t_0 / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(cosTheta * Float32(t_0 * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = t_0 / ((single(pi) * log((alpha * alpha))) * (single(1.0) + (cosTheta * (t_0 * cosTheta))));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Final simplification 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right)\right)} \]
4. Add Preprocessing

Alternative 3: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha + -1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (+ (* alpha alpha) -1.0)
  (* (- 1.0 (* cosTheta cosTheta)) (* PI (log (* alpha alpha))))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((1.0f - (cosTheta * cosTheta)) * (((float) M_PI) * logf((alpha * alpha))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(Float32(1.0) - Float32(cosTheta * cosTheta)) * Float32(Float32(pi) * log(Float32(alpha * alpha)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((single(1.0) - (cosTheta * cosTheta)) * (single(pi) * log((alpha * alpha))));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot cosTheta\right)}, cosTheta\right)\right)\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(\mathsf{neg}\left(cosTheta\right)\right), cosTheta\right)\right)\right)\right) \]

   2. neg-lowering-neg.f32 97.1%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(cosTheta\right), cosTheta\right)\right)\right)\right) \]

5. Simplified 97.1%

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

6. Final simplification 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]
7. Add Preprocessing

Alternative 4: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha + -1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\pi \cdot \left(2 \cdot \log \alpha\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (+ (* alpha alpha) -1.0)
  (* (- 1.0 (* cosTheta cosTheta)) (* PI (* 2.0 (log alpha))))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((1.0f - (cosTheta * cosTheta)) * (((float) M_PI) * (2.0f * logf(alpha))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(Float32(1.0) - Float32(cosTheta * cosTheta)) * Float32(Float32(pi) * Float32(Float32(2.0) * log(alpha)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((single(1.0) - (cosTheta * cosTheta)) * (single(pi) * (single(2.0) * log(alpha))));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) + 2 \cdot \left({\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) + {\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) + {\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right), \color{blue}{\left({\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \left(\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) + \left(cosTheta \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\pi \cdot \log \alpha\right)\right)\right)}} \]

6. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 - {cosTheta}^{2}\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\log \alpha \cdot \left(1 - {cosTheta}^{2}\right)\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \alpha \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)}\right)\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot \color{blue}{{cosTheta}^{2}}\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \log \alpha\right) \cdot \color{blue}{\left(1 + -1 \cdot {cosTheta}^{2}\right)}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right) \cdot \left(\color{blue}{1} + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right), \color{blue}{\left(1 + -1 \cdot {cosTheta}^{2}\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(2 \cdot \left(\log \alpha \cdot \mathsf{PI}\left(\right)\right)\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(\left(2 \cdot \log \alpha\right) \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{1} + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(2 \cdot \log \alpha\right), \mathsf{PI}\left(\right)\right), \left(\color{blue}{1} + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\log \alpha \cdot 2\right), \mathsf{PI}\left(\right)\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\log \alpha, 2\right), \mathsf{PI}\left(\right)\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   12. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI}\left(\right)\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   13. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI.f32}\left(\right)\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI.f32}\left(\right)\right), \left(1 + \left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)\right)\right)\right) \]

   15. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI.f32}\left(\right)\right), \left(1 - \color{blue}{{cosTheta}^{2}}\right)\right)\right) \]

   16. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{\_.f32}\left(1, \color{blue}{\left({cosTheta}^{2}\right)}\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{\_.f32}\left(1, \left(cosTheta \cdot \color{blue}{cosTheta}\right)\right)\right)\right) \]

   18. *-lowering-*.f32 97.1%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \color{blue}{cosTheta}\right)\right)\right)\right) \]

8. Simplified 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\left(\log \alpha \cdot 2\right) \cdot \pi\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)}} \]

9. Final simplification 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\pi \cdot \left(2 \cdot \log \alpha\right)\right)} \]
10. Add Preprocessing

Alternative 5: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha + -1}{\left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) \cdot \left(2 \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (+ (* alpha alpha) -1.0)
  (* (* (log alpha) (- 1.0 (* cosTheta cosTheta))) (* 2.0 PI))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((logf(alpha) * (1.0f - (cosTheta * cosTheta))) * (2.0f * ((float) M_PI)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(log(alpha) * Float32(Float32(1.0) - Float32(cosTheta * cosTheta))) * Float32(Float32(2.0) * Float32(pi))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((log(alpha) * (single(1.0) - (cosTheta * cosTheta))) * (single(2.0) * single(pi)));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\log \alpha, \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(\color{blue}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + \left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 - {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left({cosTheta}^{2}\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left(cosTheta \cdot cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{2}\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{2}\right)\right)\right) \]

   13. PI-lowering-PI.f32 97.1%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), 2\right)\right)\right) \]

5. Simplified 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) \cdot \left(\pi \cdot 2\right)}} \]

6. Final simplification 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) \cdot \left(2 \cdot \pi\right)} \]
7. Add Preprocessing

Alternative 6: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(1 + cosTheta \cdot cosTheta\right) \cdot \frac{\alpha \cdot \alpha + -1}{\log \alpha \cdot \pi}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (*
  0.5
  (*
   (+ 1.0 (* cosTheta cosTheta))
   (/ (+ (* alpha alpha) -1.0) (* (log alpha) PI)))))

C

float code(float cosTheta, float alpha) {
	return 0.5f * ((1.0f + (cosTheta * cosTheta)) * (((alpha * alpha) + -1.0f) / (logf(alpha) * ((float) M_PI))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(0.5) * Float32(Float32(Float32(1.0) + Float32(cosTheta * cosTheta)) * Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(log(alpha) * Float32(pi)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * ((single(1.0) + (cosTheta * cosTheta)) * (((alpha * alpha) + single(-1.0)) / (log(alpha) * single(pi))));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\log \alpha, \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(\color{blue}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + \left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 - {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left({cosTheta}^{2}\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left(cosTheta \cdot cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{2}\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{2}\right)\right)\right) \]

   13. PI-lowering-PI.f32 97.1%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), 2\right)\right)\right) \]

5. Simplified 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) \cdot \left(\pi \cdot 2\right)}} \]

6. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{cosTheta}^{2} \cdot \left({\alpha}^{2} - 1\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{2} \cdot \frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha}} \]
7. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{{cosTheta}^{2} \cdot \left({\alpha}^{2} - 1\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left({cosTheta}^{2} \cdot \frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{\color{blue}{{\alpha}^{2} - 1}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right) \]

   4. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\left({cosTheta}^{2} + 1\right) \cdot \color{blue}{\frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left({cosTheta}^{2} + 1\right), \color{blue}{\left(\frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left({cosTheta}^{2}\right), 1\right), \left(\frac{\color{blue}{{\alpha}^{2} - 1}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(cosTheta \cdot cosTheta\right), 1\right), \left(\frac{\color{blue}{{\alpha}^{2}} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \left(\frac{\color{blue}{{\alpha}^{2}} - 1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\left({\alpha}^{2} - 1\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)}\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\left({\alpha}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \log \alpha\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\left({\alpha}^{2} + -1\right), \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\left(-1 + {\alpha}^{2}\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \log \alpha\right)\right)\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \left({\alpha}^{2}\right)\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \log \alpha\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \left(\alpha \cdot \alpha\right)\right), \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\log \alpha}\right)\right)\right)\right) \]

   17. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \log \color{blue}{\alpha}\right)\right)\right)\right) \]

   18. log-lowering-log.f32 95.6%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right), 1\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right)\right)\right) \]

8. Simplified 95.6%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(cosTheta \cdot cosTheta + 1\right) \cdot \frac{-1 + \alpha \cdot \alpha}{\pi \cdot \log \alpha}\right)} \]

9. Final simplification 95.6%

   \[\leadsto 0.5 \cdot \left(\left(1 + cosTheta \cdot cosTheta\right) \cdot \frac{\alpha \cdot \alpha + -1}{\log \alpha \cdot \pi}\right) \]
10. Add Preprocessing

Alternative 7: 95.2% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (+ (* alpha alpha) -1.0) (* PI (log (* alpha alpha)))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / (((float) M_PI) * logf((alpha * alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(pi) * log(Float32(alpha * alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / (single(pi) * log((alpha * alpha)));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\log \left({\alpha}^{2}\right)}\right)\right) \]

   2. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \log \color{blue}{\left({\alpha}^{2}\right)}\right)\right) \]

   3. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\left({\alpha}^{2}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\left(\alpha \cdot \alpha\right)\right)\right)\right) \]

   5. *-lowering-*.f32 93.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right)\right) \]

5. Simplified 93.7%

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

6. Final simplification 93.7%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} \]
7. Add Preprocessing

Alternative 8: 67.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \alpha \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -0.5 (* (- 1.0 (* cosTheta cosTheta)) (* (log alpha) PI))))

C

float code(float cosTheta, float alpha) {
	return -0.5f / ((1.0f - (cosTheta * cosTheta)) * (logf(alpha) * ((float) M_PI)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(-0.5) / Float32(Float32(Float32(1.0) - Float32(cosTheta * cosTheta)) * Float32(log(alpha) * Float32(pi))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / ((single(1.0) - (cosTheta * cosTheta)) * (log(alpha) * single(pi)));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) + 2 \cdot \left({\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) + {\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) + {\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(2, \mathsf{+.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right), \color{blue}{\left({\alpha}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \left(\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) + \left(cosTheta \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\pi \cdot \log \alpha\right)\right)\right)}} \]

6. Taylor expanded in alpha around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 - {cosTheta}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 - {cosTheta}^{2}\right)\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \left(\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right) \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \left(\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)}\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \left(\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right) \cdot \left(1 + -1 \cdot \color{blue}{{cosTheta}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right), \color{blue}{\left(1 + -1 \cdot {cosTheta}^{2}\right)}\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI}\left(\right), \log \alpha\right), \left(\color{blue}{1} + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \log \alpha\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   8. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right), \left(1 + \left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right), \left(1 - \color{blue}{{cosTheta}^{2}}\right)\right)\right) \]

   11. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{\_.f32}\left(1, \color{blue}{\left({cosTheta}^{2}\right)}\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{\_.f32}\left(1, \left(cosTheta \cdot \color{blue}{cosTheta}\right)\right)\right)\right) \]

   13. *-lowering-*.f32 67.8%

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \color{blue}{cosTheta}\right)\right)\right)\right) \]

8. Simplified 67.8%

   \[\leadsto \color{blue}{\frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)}} \]

9. Final simplification 67.8%

   \[\leadsto \frac{-0.5}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \alpha \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 9: 65.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (/ -1.0 PI) (log (* alpha alpha))))

C

float code(float cosTheta, float alpha) {
	return (-1.0f / ((float) M_PI)) / logf((alpha * alpha));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-1.0) / Float32(pi)) / log(Float32(alpha * alpha)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) / single(pi)) / log((alpha * alpha));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\mathsf{PI}\left(\right)}}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\alpha \cdot \alpha - 1}{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\alpha \cdot \alpha - 1\right), \mathsf{PI}\left(\right)\right), \left(\color{blue}{\log \left(\alpha \cdot \alpha\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\log \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\log \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(\log \left(\color{blue}{\alpha} \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI}\left(\right)\right), \left(\log \left(\alpha \cdot \color{blue}{\alpha}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\log \left(\alpha \cdot \alpha\right), \color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)\right) \]

   11. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\alpha \cdot \alpha\right)\right), \left(\color{blue}{1} + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right), \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{+.f32}\left(1, \left(\left(\alpha \cdot \alpha - 1\right) \cdot \color{blue}{\left(cosTheta \cdot cosTheta\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(\alpha \cdot \alpha - 1\right), \color{blue}{\left(cosTheta \cdot cosTheta\right)}\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1}{\mathsf{PI}\left(\right)}\right)}, \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI}\left(\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)}, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   2. PI-lowering-PI.f32 67.9%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

7. Simplified 67.9%

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

8. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{-1}{\mathsf{PI}\left(\right)}}{\color{blue}{\log \left({\alpha}^{2}\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(1\right)}{\mathsf{PI}\left(\right)}}{\log \left({\color{blue}{\alpha}}^{2}\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}{\log \color{blue}{\left({\alpha}^{2}\right)}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \color{blue}{\log \left({\alpha}^{2}\right)}\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{PI}\left(\right)}\right), \log \color{blue}{\left({\alpha}^{2}\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{-1}{\mathsf{PI}\left(\right)}\right), \log \left({\color{blue}{\alpha}}^{2}\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI}\left(\right)\right), \log \color{blue}{\left({\alpha}^{2}\right)}\right) \]

   8. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI.f32}\left(\right)\right), \log \left({\alpha}^{\color{blue}{2}}\right)\right) \]

   9. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI.f32}\left(\right)\right), \mathsf{log.f32}\left(\left({\alpha}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI.f32}\left(\right)\right), \mathsf{log.f32}\left(\left(\alpha \cdot \alpha\right)\right)\right) \]

   11. *-lowering-*.f32 65.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI.f32}\left(\right)\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right) \]

10. Simplified 65.6%

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

Alternative 10: -0.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\pi}}{\frac{0}{0}} \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ (/ -1.0 PI) (/ 0.0 0.0)))

C

float code(float cosTheta, float alpha) {
	return (-1.0f / ((float) M_PI)) / (0.0f / 0.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-1.0) / Float32(pi)) / Float32(Float32(0.0) / Float32(0.0)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) / single(pi)) / (single(0.0) / single(0.0));
end

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-sqr-1 N/A

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   2. difference-of-sqr--1 N/A

      \[\leadsto \frac{\alpha \cdot \alpha + -1}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha + -1}{\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)}}{\color{blue}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\alpha \cdot \alpha + -1}{\mathsf{PI}\left(\right)}}{\log \left(\alpha \cdot \alpha\right)}}{\color{blue}{1} + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha + -1}{\mathsf{PI}\left(\right)}}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\alpha \cdot \alpha + -1}{\mathsf{PI}\left(\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}}{\color{blue}{\log \left(\alpha \cdot \alpha\right)}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{\alpha \cdot \alpha + -1}{\mathsf{PI}\left(\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}\right), \color{blue}{\log \left(\alpha \cdot \alpha\right)}\right) \]

4. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha \cdot \alpha + -1}{\pi}}{1 + cosTheta \cdot \left(cosTheta \cdot \left(\alpha \cdot \alpha + -1\right)\right)}}{\frac{0}{0}}} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right)}\right)}, \mathsf{/.f32}\left(0, 0\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\alpha}^{2} - 1\right), \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(\color{blue}{0}, 0\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\alpha}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\alpha}^{2} + -1\right), \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\left({\alpha}^{2}\right), -1\right), \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), -1\right), \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

   7. PI-lowering-PI.f32 -0.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

7. Simplified -0.0%

   \[\leadsto \frac{\color{blue}{\frac{\alpha \cdot \alpha + -1}{\pi}}}{\frac{0}{0}} \]

8. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1}{\mathsf{PI}\left(\right)}\right)}, \mathsf{/.f32}\left(0, 0\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI}\left(\right)\right), \mathsf{/.f32}\left(\color{blue}{0}, 0\right)\right) \]

   2. PI-lowering-PI.f32 -0.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, \mathsf{PI.f32}\left(\right)\right), \mathsf{/.f32}\left(0, 0\right)\right) \]

10. Simplified -0.0%

    \[\leadsto \frac{\color{blue}{\frac{-1}{\pi}}}{\frac{0}{0}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))