GTR1 distribution

Percentage Accurate: 98.6% → 98.6%

Time: 8.7s

Alternatives: 8

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: 98.6% 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: 98.6% 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 98.6%

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

3. Final simplification 98.6%

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

Alternative 2: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. distribute-lft1-in N/A

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

   3. +-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

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

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 3: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.2%

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 4: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate--l+ N/A

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

   2. associate-/l* N/A

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

   3. div-sub N/A

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

   4. *-lft-identity N/A

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

   5. distribute-rgt-out N/A

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

   6. *-rgt-identity N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

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

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

8. Simplified 97.2%

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

Alternative 5: 95.4% 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 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 96.3%

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

5. Simplified 96.3%

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

6. Final simplification 96.3%

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

Alternative 6: 66.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

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

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

   11. associate-*l* N/A

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

   12. cube-unmult N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(\alpha \cdot {\alpha}^{3}\right)\right), \left(\left(\mathsf{neg}\left(1\right)\right) - \alpha \cdot \alpha\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), cosTheta\right), cosTheta\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\alpha, \left({\alpha}^{3}\right)\right)\right), \left(\left(\mathsf{neg}\left(1\right)\right) - \alpha \cdot \alpha\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), cosTheta\right), cosTheta\right)\right)\right)\right) \]

   14. cube-unmult N/A

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

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

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

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

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

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

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

   18. metadata-eval N/A

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

   19. *-lowering-*.f32 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in alpha around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 67.2%

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

7. Simplified 67.2%

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

8. Final simplification 67.2%

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

Alternative 7: 65.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\pi \cdot \log \alpha} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

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

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

   9. metadata-eval 98.2%

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

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

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

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

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

   4. distribute-rgt-out N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f32 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

   4. log-lowering-log.f32 66.0%

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

10. Simplified 66.0%

    \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \log \alpha}} \]
11. Add Preprocessing

Alternative 8: -0.0% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

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

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

4. Applied egg-rr -0.0%

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

5. Taylor expanded in alpha around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 -0.0%

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

7. Simplified -0.0%

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

8. Taylor expanded in alpha around 0

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

10. Simplified -0.0%

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

11. Taylor expanded in cosTheta around 0

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

13. Simplified -0.0%

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

Reproduce

herbie shell --seed 2024145 
(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)))))