GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 26.6s

Alternatives: 12

Speedup: 1.0×

• unsoundness detected in the egraph

Specification

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

Math

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

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

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

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: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (((alpha * alpha) - single(0.5)) - single(0.5)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((single(-1.0) * 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. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. metadata-eval N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 97.6%

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

6. Applied rewrites 97.6%

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

Alternative 2: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (2.0f + 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))) * Float32(Float32(2.0) + 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 0

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

   1. +-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 97.6%

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

   6. lift-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(3 - \left(3 - \left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta\right)\right)\right)} \]

   8. lower-neg.f32 97.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(3 - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)\right)} \]

6. Applied rewrites 97.6%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+r- N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(\color{blue}{4} - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(\color{blue}{\left(2 + 2\right)} - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)} \]

   6. associate--l+ N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\left(2 + \left(2 - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(\color{blue}{\left(-1 + 3\right)} - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)\right)} \]

   8. associate-+r- N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \color{blue}{\left(-1 + \left(3 - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)\right)}\right)} \]

   9. lift--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \color{blue}{\left(3 - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)}\right)\right)} \]

   10. lift--.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \color{blue}{\left(3 - \left(3 - \left(-cosTheta\right) \cdot cosTheta\right)\right)}\right)\right)} \]

   11. lift--.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \left(3 - \color{blue}{\left(3 - \left(-cosTheta\right) \cdot cosTheta\right)}\right)\right)\right)} \]

   12. associate--r- N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \color{blue}{\left(\left(3 - 3\right) + \left(-cosTheta\right) \cdot cosTheta\right)}\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \left(\color{blue}{0} + \left(-cosTheta\right) \cdot cosTheta\right)\right)\right)} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \left(0 + \color{blue}{\left(-cosTheta\right) \cdot cosTheta}\right)\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(2 + \left(-1 + \left(0 + \color{blue}{cosTheta \cdot \left(-cosTheta\right)}\right)\right)\right)} \]

8. Applied rewrites 97.6%

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

Alternative 3: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((single(-1.0) * 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. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

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(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

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 97.6%

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

   5. lift-*.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 97.6%

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

6. Applied rewrites 97.6%

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

Alternative 5: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (((alpha * alpha) - single(0.5)) - single(0.5)) / ((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 95.6%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. metadata-eval N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 95.5%

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

6. Applied rewrites 95.5%

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

Alternative 6: 95.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - 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 95.6%

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

Alternative 7: 95.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * 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 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 95.6%

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

   1. --rgt-identity N/A

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

   2. lift--.f32 N/A

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

   3. lift--.f32 N/A

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

   4. metadata-eval N/A

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

   5. associate--r- N/A

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

   6. lower-+.f32 N/A

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

   7. lower--.f32 94.6%

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

   8. lift--.f32 N/A

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

   9. sub-flip N/A

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

   10. lift-*.f32 N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 94.6%

       \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)} - 3\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

6. Applied rewrites 94.6%

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

   1. lift-+.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) - 3\right) + 3}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   2. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) - 3\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   3. lift-fma.f32 N/A

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

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

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

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

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-+l- N/A

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

   9. metadata-eval N/A

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

   10. --rgt-identity 95.6%

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

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

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

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

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

   15. lift-fma.f32 95.5%

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

8. Applied rewrites 95.5%

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

Alternative 8: 66.0% accurate, 2.0× speedup

Math

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

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 95.6%

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

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

Alternative 9: 3.1% accurate, 2.0× speedup

Math

\[\frac{-1}{\left(\pi \cdot \left(-\left(-0.5 - \mathsf{fma}\left(\alpha, \alpha, -0.5\right)\right)\right)\right) \cdot 1} \]

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(-1.0) / Float32(Float32(Float32(pi) * Float32(-Float32(Float32(-0.5) - fma(alpha, alpha, Float32(-0.5))))) * 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 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 95.6%

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

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-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. *-rgt-identity N/A

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

   9. metadata-eval N/A

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

   10. mul-fabs N/A

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

   11. lower-fabs.f32 N/A

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

   12. *-rgt-identity 66.0%

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

9. Applied rewrites 66.0%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. count-2-rev N/A

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

    4. flip-+ N/A

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

    5. +-inverses N/A

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

    6. +-inverses N/A

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

    7. lower-/.f32 -0.0%

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

    8. lower-/.f32 N/A

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

    9. +-inverses N/A

       \[\leadsto \frac{-1}{\left(\pi \cdot \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{1}{2}\right) - \left(\left(\alpha \cdot \alpha\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{1}{2}\right)}}{0}\right) \cdot 1} \]

    10. +-inverses N/A

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

    11. flip-+ N/A

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

    12. distribute-lft-out N/A

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

    13. metadata-eval N/A

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

    14. *-rgt-identity 3.0%

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

    15. +-rgt-identity N/A

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

    16. metadata-eval N/A

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

    17. associate--l+ N/A

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

    18. lift-*.f32 N/A

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

    19. lift-fma.f32 N/A

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

    20. sub-negate-rev N/A

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

    21. lower-neg.f32 N/A

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

    22. lower--.f32 3.1%

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

11. Applied rewrites 3.1%

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

Alternative 10: 3.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(Float32(1.0) * Float32(pi)) * Float32(alpha * alpha)))
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 95.6%

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

   1. --rgt-identity N/A

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

   2. lift--.f32 N/A

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

   3. lift--.f32 N/A

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

   4. metadata-eval N/A

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

   5. associate--r- N/A

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

   6. lower-+.f32 N/A

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

   7. lower--.f32 94.6%

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

   8. lift--.f32 N/A

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

   9. sub-flip N/A

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

   10. lift-*.f32 N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f32 94.6%

       \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)} - 3\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

6. Applied rewrites 94.6%

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

   1. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) - 3\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   2. lift-fma.f32 N/A

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

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

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

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

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

   5. lift-*.f32 N/A

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

   6. associate--l- N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha \cdot \alpha - \color{blue}{4}\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\left(\alpha \cdot \alpha - \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)}\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{\left(\left(\alpha \cdot 1\right) \cdot \left(\alpha \cdot 1\right) - \left(\mathsf{neg}\left(\color{blue}{-4}\right)\right)\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\left(\left(\alpha \cdot 1\right) \cdot \left(\alpha \cdot 1\right) - \color{blue}{4}\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\left(\left(\alpha \cdot 1\right) \cdot \left(\alpha \cdot 1\right) - \color{blue}{2 \cdot 2}\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   16. difference-of-squares N/A

       \[\leadsto \frac{\color{blue}{\left(\alpha \cdot 1 + 2\right) \cdot \left(\alpha \cdot 1 - 2\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{\left(\alpha \cdot 1 + 2\right) \cdot \left(\alpha \cdot 1 - 2\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   18. lower-+.f32 N/A

       \[\leadsto \frac{\color{blue}{\left(\alpha \cdot 1 + 2\right)} \cdot \left(\alpha \cdot 1 - 2\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   19. *-rgt-identity N/A

       \[\leadsto \frac{\left(\color{blue}{\alpha} + 2\right) \cdot \left(\alpha \cdot 1 - 2\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   20. lower--.f32 N/A

       \[\leadsto \frac{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha \cdot 1 - 2\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   21. *-rgt-identity 93.3%

       \[\leadsto \frac{\left(\alpha + 2\right) \cdot \left(\color{blue}{\alpha} - 2\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

8. Applied rewrites 93.3%

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

   1. lift-+.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha - 2\right) + 3}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\alpha + 2\right) \cdot \left(\alpha - 2\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   3. lift-+.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\alpha + 2\right)} \cdot \left(\alpha - 2\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   4. lift--.f32 N/A

      \[\leadsto \frac{\left(\alpha + 2\right) \cdot \color{blue}{\left(\alpha - 2\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   5. difference-of-squares-rev N/A

      \[\leadsto \frac{\color{blue}{\left(\alpha \cdot \alpha - 2 \cdot 2\right)} + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\left(\color{blue}{\alpha \cdot \alpha} - 2 \cdot 2\right) + 3}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   7. associate-+l- N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha - \left(2 \cdot 2 - 3\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha} - \left(2 \cdot 2 - 3\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - \left(\color{blue}{4} - 3\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   10. metadata-eval N/A

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

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

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

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

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

   13. lower-fma.f32 95.5%

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f32 N/A

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

10. Applied rewrites 3.0%

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

Alternative 11: 3.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-1.0) / (((alpha * alpha) * single(pi)) * 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 95.6%

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

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-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. *-rgt-identity N/A

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

   9. metadata-eval N/A

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

   10. mul-fabs N/A

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

   11. lower-fabs.f32 N/A

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

   12. *-rgt-identity 66.0%

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

9. Applied rewrites 66.0%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. lower-log.f32 N/A

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

    8. lift-fabs.f32 N/A

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

    9. log-pow N/A

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

    10. pow2 N/A

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

    11. lift-*.f32 N/A

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

    12. lift-log.f32 N/A

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

    13. lower-*.f32 66.0%

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

11. Applied rewrites 3.0%

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

Reproduce

herbie shell --seed 2025322 
(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)))))