GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 3.3s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))) * fma(t_0, 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f32 N/A

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

   11. pow2 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 98.5

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\pi \cdot \left(\left(\log \alpha \cdot 2\right) \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, t\_0, 1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/ t_0 (* PI (* (* (log alpha) 2.0) (fma (* cosTheta cosTheta) t_0 1.0))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / (((float) M_PI) * ((logf(alpha) * 2.0f) * fmaf((cosTheta * cosTheta), t_0, 1.0f)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	return Float32(t_0 / Float32(Float32(pi) * Float32(Float32(log(alpha) * Float32(2.0)) * fma(Float32(cosTheta * cosTheta), t_0, 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. 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. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. pow2 N/A

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

   6. lower-log.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.5%

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

Alternative 4: 97.5% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (/ (- (* alpha alpha) 1.0) PI)
  (* (* (log 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) * 2.0f) * fmaf(-cosTheta, cosTheta, 1.0f));
}

Julia

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 98.7%

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 97.4

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

8. Applied rewrites 97.4%

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

Alternative 5: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f32 N/A

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

   11. pow2 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 98.5

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. mul-1-neg N/A

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

   10. +-commutative N/A

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

   11. lower-+.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 97.5

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

6. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

8. Applied rewrites 97.5%

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

Alternative 6: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / (((log(alpha) * single(pi)) * (single(1.0) - (cosTheta * cosTheta))) * single(2.0));
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f32 N/A

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

   11. pow2 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 98.5

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. mul-1-neg N/A

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

   10. +-commutative N/A

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

   11. lower-+.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 97.5

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower--.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 97.5

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

9. Applied rewrites 97.5%

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

Alternative 7: 95.3% accurate, 1.6× speedup

Math

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

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 95.3%

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

Alternative 8: 95.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (((alpha * alpha) - single(1.0)) / single(pi)) / (log(alpha) * single(2.0));
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. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 98.7%

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\color{blue}{2 \cdot \log \alpha}} \]
7. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   3. lift--.f32 N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   7. distribute-lft-in N/A

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

   8. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   9. lift--.f32 N/A

      \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{2 \cdot \log \alpha} \]

   10. *-commutative N/A

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

   11. lift-log.f32 N/A

       \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\log \alpha \cdot 2} \]

   12. lift-*.f32 95.2

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

8. Applied rewrites 95.2%

   \[\leadsto \frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\color{blue}{\log \alpha \cdot 2}} \]
9. Add Preprocessing

Alternative 9: 95.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. log-pow-rev N/A

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

   2. pow-pow N/A

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

   3. log-pow-rev N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. lift-PI.f32 95.3

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

4. Applied rewrites 95.3%

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

Alternative 10: 65.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.3%

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

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 65.6%

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

Alternative 11: 65.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-1.0) / (((log(alpha) * single(2.0)) * 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 95.3%

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

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 65.6%

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

   1. pow2 65.6

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

   2. remove-double-div 65.6

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

   3. pow2 65.6

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

   4. pow2 N/A

      \[\leadsto \frac{-1}{\left(\mathsf{Rewrite=>}\left(lift-PI.f32, \mathsf{PI}\left(\right)\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

   5. pow2 N/A

      \[\leadsto \frac{-1}{\mathsf{Rewrite=>}\left(lift-*.f32, \left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)\right) \cdot 1} \]

   6. pow2 N/A

      \[\leadsto \frac{-1}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{Rewrite=>}\left(lift-log.f32, \log \left(\alpha \cdot \alpha\right)\right)\right) \cdot 1} \]

   7. pow2 N/A

      \[\leadsto \frac{-1}{\left(\mathsf{PI}\left(\right) \cdot \log \mathsf{Rewrite=>}\left(lift-*.f32, \left(\alpha \cdot \alpha\right)\right)\right) \cdot 1} \]

   8. pow2 N/A

      \[\leadsto \frac{-1}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{Rewrite=>}\left(log-prod, \left(\log \alpha + \log \alpha\right)\right)\right) \cdot 1} \]

   9. pow2 N/A

      \[\leadsto \frac{-1}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{Rewrite<=}\left(count-2-rev, \left(2 \cdot \log \alpha\right)\right)\right) \cdot 1} \]

   10. pow2 N/A

       \[\leadsto \frac{-1}{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(2 \cdot \log \alpha\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 1} \]

   11. pow2 N/A

       \[\leadsto \frac{-1}{\mathsf{Rewrite=>}\left(lower-*.f32, \left(\left(2 \cdot \log \alpha\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 1} \]

   12. pow2 N/A

       \[\leadsto \frac{-1}{\left(\mathsf{Rewrite=>}\left(*-commutative, \left(\log \alpha \cdot 2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 1} \]

   13. pow2 N/A

       \[\leadsto \frac{-1}{\left(\left(\mathsf{Rewrite<=}\left(lift-log.f32, \log \alpha\right) \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 1} \]

   14. pow2 N/A

       \[\leadsto \frac{-1}{\left(\mathsf{Rewrite<=}\left(lift-*.f32, \left(\log \alpha \cdot 2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 1} \]

   15. pow2 N/A

       \[\leadsto \frac{-1}{\left(\left(\log \alpha \cdot 2\right) \cdot \mathsf{Rewrite<=}\left(lift-PI.f32, \pi\right)\right) \cdot 1} \]

9. Applied rewrites 65.6%

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

Reproduce

herbie shell --seed 2025117 
(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)))))