GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 5.3s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

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: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 98.5%

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 98.5%

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

   14. lower-unsound-log.f32 N/A

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

   15. lower-unsound-*.f32 N/A

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

   16. lower-unsound-exp.f32 N/A

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

   17. pow-to-exp N/A

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

   18. lower-pow.f32 98.7%

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-flip N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. mul-1-neg N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 98.5%

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-flip N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. mul-1-neg N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 98.5%

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

3. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.5%

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. add-flip N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 98.5%

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

   9. lift-+.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. sub-flip-reverse N/A

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

   12. lower--.f32 98.5%

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 98.5%

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

5. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 98.5%

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

7. Applied rewrites 98.5%

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

Alternative 4: 97.6% accurate, 1.1× 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 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. pow2 N/A

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

   6. log-pow-rev N/A

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

   7. lift-log.f32 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f32 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f32 N/A

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

   14. lower-*.f32 97.5%

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

   15. lift-+.f32 N/A

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

6. Applied rewrites 97.5%

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

Alternative 6: 95.4% accurate, 1.2× 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 95.4%

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

Alternative 7: 95.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) + single(pi)) * log(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 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-unsound-log.f32 95.4%

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-log.f32 95.4%

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

9. Applied rewrites 95.4%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f32 N/A

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

    5. count-2-rev N/A

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

    6. lower-+.f32 95.4%

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

11. Applied rewrites 95.4%

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

Alternative 8: 65.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-1.0) / (single(2.0) * (single(pi) * log(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 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-unsound-log.f32 95.4%

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-log.f32 95.4%

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

9. Applied rewrites 95.4%

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

10. Taylor expanded in alpha around 0

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

12. Applied rewrites 65.4%

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

Reproduce

herbie shell --seed 2025258 
(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)))))