GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 8.1s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 98.6

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 98.7

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

10. Applied rewrites 98.7%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * Float32(Float32(pi) * log(Float32(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. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 98.6

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(Float32(pi) * log(alpha)) * fma(Float32(cosTheta * cosTheta), Float32(-2.0), Float32(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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 98.6

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 97.7

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

10. Applied rewrites 97.7%

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

Alternative 4: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. flip3-- N/A

      \[\leadsto \frac{\color{blue}{\frac{{\left(\alpha \cdot \alpha\right)}^{3} - {1}^{3}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + \left(1 \cdot 1 + \left(\alpha \cdot \alpha\right) \cdot 1\right)}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. clear-num N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + \left(1 \cdot 1 + \left(\alpha \cdot \alpha\right) \cdot 1\right)}{{\left(\alpha \cdot \alpha\right)}^{3} - {1}^{3}}}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + \left(1 \cdot 1 + \left(\alpha \cdot \alpha\right) \cdot 1\right)}{{\left(\alpha \cdot \alpha\right)}^{3} - {1}^{3}}}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. clear-num N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{\frac{{\left(\alpha \cdot \alpha\right)}^{3} - {1}^{3}}{\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) + \left(1 \cdot 1 + \left(\alpha \cdot \alpha\right) \cdot 1\right)}}}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. flip3-- N/A

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

   7. lift--.f32 N/A

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

   8. lower-/.f32 98.0

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

   9. lift--.f32 N/A

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

   10. sub-neg N/A

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

   11. lift-*.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 97.1%

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

8. Taylor expanded in cosTheta around 0

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

   1. associate--l+ N/A

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

   2. div-sub N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. associate-/l* N/A

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

   7. distribute-rgt-out-- N/A

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

   8. sub-neg N/A

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

10. Applied rewrites 97.0%

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

Alternative 5: 95.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 98.6

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 98.7

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

10. Applied rewrites 98.7%

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

11. Taylor expanded in cosTheta around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 N/A

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

    4. lower-log.f32 96.2

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

13. Applied rewrites 96.2%

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

Alternative 6: 95.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 98.6

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.6

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 98.7

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

10. Applied rewrites 98.7%

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

11. Taylor expanded in cosTheta around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f32 N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. distribute-lft-in N/A

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

    6. metadata-eval N/A

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

    7. lower-fma.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-*.f32 N/A

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

    11. lower-PI.f32 N/A

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

    12. lower-log.f32 96.1

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

13. Applied rewrites 96.1%

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

Alternative 7: 65.7% accurate, 1.3× speedup

Math

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

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(-1.0) / Float32(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. Taylor expanded in alpha around 0

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 63.0

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

8. Applied rewrites 63.0%

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

Alternative 8: -0.0% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-+.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. remove-double-neg N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in alpha around 0

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \color{blue}{\left(-1 \cdot cosTheta\right)}, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites -0.0%

    \[\leadsto \frac{\color{blue}{-1}}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(-cosTheta\right), \pi, \pi\right)} \]

11. Taylor expanded in cosTheta around 0

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

    1. lower-PI.f32 -0.0

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

13. Applied rewrites -0.0%

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

14. Final simplification -0.0%

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

Reproduce

herbie shell --seed 2024219 
(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)))))