GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 16.9s

Alternatives: 5

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ t_1 := {\left(\alpha \cdot \alpha\right)}^{\pi}\\ \frac{t\_0}{\log \left({t\_1}^{\left(\left(cosTheta \cdot cosTheta\right) \cdot t\_0\right)} \cdot {t\_1}^{1}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)) (t_1 (pow (* alpha alpha) PI)))
   (/ t_0 (log (* (pow t_1 (* (* cosTheta cosTheta) t_0)) (pow t_1 1.0))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	float t_1 = powf((alpha * alpha), ((float) M_PI));
	return t_0 / logf((powf(t_1, ((cosTheta * cosTheta) * t_0)) * powf(t_1, 1.0f)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	t_1 = Float32(alpha * alpha) ^ Float32(pi)
	return Float32(t_0 / log(Float32((t_1 ^ Float32(Float32(cosTheta * cosTheta) * t_0)) * (t_1 ^ Float32(1.0)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) - single(1.0);
	t_1 = (alpha * alpha) ^ single(pi);
	tmp = t_0 / log(((t_1 ^ ((cosTheta * cosTheta) * t_0)) * (t_1 ^ single(1.0))));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   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-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. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. log-pow-rev N/A

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

4. Applied rewrites 98.7%

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

   1. lift-pow.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. unpow-prod-up N/A

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

   10. lower-*.f32 N/A

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

6. Applied rewrites 98.7%

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

Alternative 2: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(cosTheta \cdot cosTheta, \alpha \cdot \alpha - 1, 1\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)\\ t_1 := {t\_0}^{2}\\ \frac{\left(\alpha \cdot \alpha\right) \cdot t\_0}{t\_1} - \frac{t\_0}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0
         (*
          (fma (* cosTheta cosTheta) (- (* alpha alpha) 1.0) 1.0)
          (log (pow (* alpha alpha) PI))))
        (t_1 (pow t_0 2.0)))
   (- (/ (* (* alpha alpha) t_0) t_1) (/ t_0 t_1))))

C

float code(float cosTheta, float alpha) {
	float t_0 = fmaf((cosTheta * cosTheta), ((alpha * alpha) - 1.0f), 1.0f) * logf(powf((alpha * alpha), ((float) M_PI)));
	float t_1 = powf(t_0, 2.0f);
	return (((alpha * alpha) * t_0) / t_1) - (t_0 / t_1);
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(fma(Float32(cosTheta * cosTheta), Float32(Float32(alpha * alpha) - Float32(1.0)), Float32(1.0)) * log((Float32(alpha * alpha) ^ Float32(pi))))
	t_1 = t_0 ^ Float32(2.0)
	return Float32(Float32(Float32(Float32(alpha * alpha) * t_0) / t_1) - Float32(t_0 / t_1))
end

Derivation

1. Initial program 98.6%

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

4. Applied rewrites 98.6%

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

6. Applied rewrites 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 98.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

4. Applied rewrites 98.3%

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

Alternative 4: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{{\alpha}^{4} - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(\left(1 + \alpha \cdot \alpha\right) \cdot \left(\left(1 + {\left(\alpha \cdot cosTheta\right)}^{2}\right) - cosTheta \cdot cosTheta\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (pow alpha 4.0) 1.0)
  (*
   (log (pow (* alpha alpha) PI))
   (*
    (+ 1.0 (* alpha alpha))
    (- (+ 1.0 (pow (* alpha cosTheta) 2.0)) (* cosTheta cosTheta))))))

C

float code(float cosTheta, float alpha) {
	return (powf(alpha, 4.0f) - 1.0f) / (logf(powf((alpha * alpha), ((float) M_PI))) * ((1.0f + (alpha * alpha)) * ((1.0f + powf((alpha * cosTheta), 2.0f)) - (cosTheta * cosTheta))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32((alpha ^ Float32(4.0)) - Float32(1.0)) / Float32(log((Float32(alpha * alpha) ^ Float32(pi))) * Float32(Float32(Float32(1.0) + Float32(alpha * alpha)) * Float32(Float32(Float32(1.0) + (Float32(alpha * cosTheta) ^ Float32(2.0))) - Float32(cosTheta * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha ^ single(4.0)) - single(1.0)) / (log(((alpha * alpha) ^ single(pi))) * ((single(1.0) + (alpha * alpha)) * ((single(1.0) + ((alpha * cosTheta) ^ single(2.0))) - (cosTheta * cosTheta))));
end

Derivation

1. Initial program 98.6%

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

   1. lift-*.f32 N/A

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

   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-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. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. log-pow-rev N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in alpha around inf

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

   1. pow2 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. sqr-pow N/A

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

   6. metadata-eval N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. pow-flip N/A

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

   13. metadata-eval N/A

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

   14. lower-pow.f32 98.3

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in cosTheta around inf

   \[\leadsto \color{blue}{\frac{{\alpha}^{4} - 1}{\log \left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)}} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\color{blue}{\log \left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)}} \]

   2. lift-pow.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \color{blue}{\left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\color{blue}{\log \left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)}} \]

   5. lower-log.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \left(\color{blue}{\left(1 + {\alpha}^{2}\right)} \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   6. pow2 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   8. lift-pow.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\color{blue}{1} + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{{\alpha}^{4} - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(\left(1 + {\alpha}^{2}\right) \cdot \color{blue}{\left(\left(1 + {\alpha}^{2} \cdot {cosTheta}^{2}\right) - {cosTheta}^{2}\right)}\right)} \]

10. Applied rewrites 98.3%

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

Alternative 5: 98.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \alpha \cdot \pi\\ t_1 := \mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\\ t_2 := \frac{0.5}{t\_0 \cdot t\_1}\\ t_3 := t\_2 - \frac{-0.5 \cdot \left(cosTheta \cdot cosTheta\right)}{t\_0 \cdot {t\_1}^{2}}\\ \mathsf{fma}\left(\frac{{\left(cosTheta \cdot \alpha\right)}^{2} \cdot t\_3}{t\_1}, -1, t\_3\right) \cdot \left(\alpha \cdot \alpha\right) - t\_2 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (* (log alpha) PI))
        (t_1 (fma (* cosTheta cosTheta) -1.0 1.0))
        (t_2 (/ 0.5 (* t_0 t_1)))
        (t_3 (- t_2 (/ (* -0.5 (* cosTheta cosTheta)) (* t_0 (pow t_1 2.0))))))
   (-
    (*
     (fma (/ (* (pow (* cosTheta alpha) 2.0) t_3) t_1) -1.0 t_3)
     (* alpha alpha))
    t_2)))

C

float code(float cosTheta, float alpha) {
	float t_0 = logf(alpha) * ((float) M_PI);
	float t_1 = fmaf((cosTheta * cosTheta), -1.0f, 1.0f);
	float t_2 = 0.5f / (t_0 * t_1);
	float t_3 = t_2 - ((-0.5f * (cosTheta * cosTheta)) / (t_0 * powf(t_1, 2.0f)));
	return (fmaf(((powf((cosTheta * alpha), 2.0f) * t_3) / t_1), -1.0f, t_3) * (alpha * alpha)) - t_2;
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(log(alpha) * Float32(pi))
	t_1 = fma(Float32(cosTheta * cosTheta), Float32(-1.0), Float32(1.0))
	t_2 = Float32(Float32(0.5) / Float32(t_0 * t_1))
	t_3 = Float32(t_2 - Float32(Float32(Float32(-0.5) * Float32(cosTheta * cosTheta)) / Float32(t_0 * (t_1 ^ Float32(2.0)))))
	return Float32(Float32(fma(Float32(Float32((Float32(cosTheta * alpha) ^ Float32(2.0)) * t_3) / t_1), Float32(-1.0), t_3) * Float32(alpha * alpha)) - t_2)
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(cosTheta \cdot \alpha\right)}^{2} \cdot \left(\frac{0.5}{\left(\log \alpha \cdot \pi\right) \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5 \cdot \left(cosTheta \cdot cosTheta\right)}{\left(\log \alpha \cdot \pi\right) \cdot {\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\left(\log \alpha \cdot \pi\right) \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5 \cdot \left(cosTheta \cdot cosTheta\right)}{\left(\log \alpha \cdot \pi\right) \cdot {\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right) - \frac{0.5}{\left(\log \alpha \cdot \pi\right) \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))