GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 10.0s

Alternatives: 9

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, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + (cosTheta * (((alpha * alpha) * 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 * Float32(Float32(Float32(alpha * alpha) * cosTheta) - cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + (cosTheta * (((alpha * alpha) * 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. Step-by-step derivation

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

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

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

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

   3. *-commutative N/A

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

   4. distribute-rgt-in N/A

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

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

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

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

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

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

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

   8. neg-mul-1 N/A

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

   9. neg-lowering-neg.f32 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta - cosTheta\right)\right)} \]
6. 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 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 3: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((single(pi) * log(alpha)) * ((single(1.0) - (cosTheta * cosTheta)) * 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 97.9%

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

5. Simplified 97.9%

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

6. Final simplification 97.9%

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

Alternative 4: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * ((single(1.0) + (cosTheta * cosTheta)) * (((alpha * alpha) + single(-1.0)) / (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. 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 97.9%

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

5. Simplified 97.9%

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

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. +-lowering-+.f32 N/A

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

   13. unpow2 N/A

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

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

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

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

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

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

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

   17. log-lowering-log.f32 97.3%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

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

Alternative 5: 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 98.6%

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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / (log(alpha) * (single(pi) * (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. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

   15. associate-*l* N/A

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

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

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

3. Simplified 98.5%

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

   1. remove-double-neg N/A

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

   2. neg-log N/A

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

   3. neg-log N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f32 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in alpha around 0

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

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 66.8%

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

9. Simplified 66.8%

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

Alternative 7: 65.5% 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 98.6%

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

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

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

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

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

   3. *-commutative N/A

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

   4. distribute-rgt-in N/A

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

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

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

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

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

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

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

   8. neg-mul-1 N/A

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

   9. neg-lowering-neg.f32 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f32 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in alpha around 0

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

10. Simplified 65.9%

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

Alternative 8: 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. 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 97.9%

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

5. Simplified 97.9%

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

6. Taylor expanded in alpha around 0

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

8. Simplified 66.8%

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

9. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}} \]
10. 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 65.9%

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

11. Simplified 65.9%

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

Alternative 9: -0.0% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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-/l/ N/A

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

   5. associate-/l/ N/A

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

   6. associate-/r* N/A

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

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

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

4. Applied egg-rr -0.0%

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

5. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 -0.0%

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

7. Simplified -0.0%

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

8. Taylor expanded in cosTheta around 0

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

10. Simplified -0.0%

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

11. Taylor expanded in alpha around 0

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

13. Simplified -0.0%

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

14. Final simplification -0.0%

    \[\leadsto \frac{\frac{-1}{\frac{0}{0}}}{\pi} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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)))))