GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 5.0s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / (logf(powf((alpha * alpha), ((float) M_PI))) * (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(log((Float32(alpha * alpha) ^ Float32(pi))) * 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 / (log(((alpha * alpha) ^ single(pi))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Derivation

1. Initial program 98.5%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 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. lift-log.f32 N/A

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

   4. log-pow-rev N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 98.5

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

4. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. mul-1-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \mathsf{fma}\left(\alpha \cdot \alpha, cosTheta, \mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta\right)} \]

   6. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 98.5

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

6. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha \cdot \alpha, cosTheta, -cosTheta\right), cosTheta, 1\right)}} \]
7. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. mul-1-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \mathsf{fma}\left(\alpha \cdot \alpha, cosTheta, \mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta\right)} \]

   6. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-log.f32 98.5

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

   14. lift-+.f32 N/A

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

6. Applied rewrites 98.6%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\alpha \cdot \alpha, cosTheta, -cosTheta\right), cosTheta, 1\right)\right)}} \]
7. Add Preprocessing

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. pow2 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta, cosTheta, 1\right)\right)} \]

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

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

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

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

   6. lower-fma.f32 98.5

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right)\right)} \]

9. Applied rewrites 98.5%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(cosTheta, \mathsf{neg}\left(cosTheta\right), 1\right)} \]

   8. lift-neg.f32 97.6

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(cosTheta, -cosTheta, 1\right)} \]

7. Applied rewrites 97.6%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(cosTheta, \color{blue}{-cosTheta}, 1\right)} \]
8. Add Preprocessing

Alternative 7: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.1%

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

7. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(\mathsf{neg}\left(cosTheta\right), cosTheta, 1\right)\right)} \]

   7. lift-neg.f32 97.6

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-cosTheta, cosTheta, 1\right)\right)} \]

9. Applied rewrites 97.6%

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

Alternative 8: 95.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.4%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 66.0%

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
8. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. log-pow-rev N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

   8. log-pow-rev N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-log.f32 N/A

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

   15. lift-PI.f32 66.0

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

9. Applied rewrites 66.0%

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

10. Taylor expanded in cosTheta around 0

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

12. Applied rewrites 95.4%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot \pi}} \]
13. Add Preprocessing

Alternative 9: 66.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.4%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 66.0%

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
8. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. log-pow-rev N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

   8. log-pow-rev N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-log.f32 N/A

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

   15. lift-PI.f32 66.0

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

9. Applied rewrites 66.0%

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

Alternative 10: 66.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.4%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 66.0%

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

Reproduce

herbie shell --seed 2025093 
(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)))))