GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 5.9s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (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

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (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, 0.8× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (- (* alpha alpha) 1.0)
 (log
  (pow
   (* alpha alpha)
   (* (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0) PI)))))

C

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

Julia

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (/
  (fma alpha alpha -1.0)
  (*
   (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0)
   (log (* alpha alpha))))
 PI))

C

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

Julia

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

3. Applied rewrites 98.4%

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

Alternative 3: 98.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (- (* alpha alpha) 1.0)
 (*
  (* (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0) PI)
  (log (* alpha alpha)))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.7%

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

5. Applied rewrites 98.5%

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

Alternative 4: 98.3% accurate, 1.0× speedup

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

Math

\[\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \left(2 \cdot \log \alpha\right)} \cdot 0.31830987334251404 \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (*
 (/
  (fma alpha alpha -1.0)
  (*
   (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0)
   (* 2.0 (log alpha))))
 0.31830987334251404))

C

float code(float cosTheta, float alpha) {
	return (fmaf(alpha, alpha, -1.0f) / (fmaf((cosTheta * cosTheta), fmaf(alpha, alpha, -1.0f), 1.0f) * (2.0f * logf(alpha)))) * 0.31830987334251404f;
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(fma(Float32(cosTheta * cosTheta), fma(alpha, alpha, Float32(-1.0)), Float32(1.0)) * Float32(Float32(2.0) * log(alpha)))) * Float32(0.31830987334251404))
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

3. Applied rewrites 98.2%

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

4. Evaluated real constant 98.2%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \log \left(\alpha \cdot \alpha\right)} \cdot 0.31830987334251404 \]
5. Step-by-step derivation

6. Applied rewrites 98.3%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \left(2 \cdot \log \alpha\right)} \cdot 0.31830987334251404 \]
7. Add Preprocessing

Alternative 5: 98.2% accurate, 1.0× speedup

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

Math

\[\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \log \left(\alpha \cdot \alpha\right)} \cdot 0.31830987334251404 \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (*
 (/
  (fma alpha alpha -1.0)
  (*
   (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0)
   (log (* alpha alpha))))
 0.31830987334251404))

C

float code(float cosTheta, float alpha) {
	return (fmaf(alpha, alpha, -1.0f) / (fmaf((cosTheta * cosTheta), fmaf(alpha, alpha, -1.0f), 1.0f) * logf((alpha * alpha)))) * 0.31830987334251404f;
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(fma(Float32(cosTheta * cosTheta), fma(alpha, alpha, Float32(-1.0)), Float32(1.0)) * log(Float32(alpha * alpha)))) * Float32(0.31830987334251404))
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

3. Applied rewrites 98.2%

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

4. Evaluated real constant 98.2%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \log \left(\alpha \cdot \alpha\right)} \cdot 0.31830987334251404 \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 1.3× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (/ (fma alpha alpha -1.0) (log (* alpha alpha)))
 (* PI (fma (- cosTheta) cosTheta 1.0))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.0%

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

8. Applied rewrites 97.4%

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

Alternative 7: 97.4% accurate, 1.3× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (fma alpha alpha -1.0)
 (* (* (fma (- cosTheta) cosTheta 1.0) (log (* alpha alpha))) PI)))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.0%

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

8. Applied rewrites 96.9%

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

10. Applied rewrites 97.4%

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

Alternative 8: 97.4% accurate, 1.4× speedup

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

Math

\[\frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(6.2831854820251465 \cdot \log \alpha\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (- (* alpha alpha) 1.0)
 (*
  (fma (- cosTheta) cosTheta 1.0)
  (* 6.2831854820251465 (log alpha)))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / (fmaf(-cosTheta, cosTheta, 1.0f) * (6.2831854820251465f * logf(alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(fma(Float32(-cosTheta), cosTheta, Float32(1.0)) * Float32(Float32(6.2831854820251465) * log(alpha))))
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 97.4%

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

9. Applied rewrites 97.4%

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

10. Evaluated real constant 97.4%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(6.2831854820251465 \cdot \log \alpha\right)} \]
11. Add Preprocessing

Alternative 9: 95.2% accurate, 1.6× speedup

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

Math

\[\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -0.5\right) - 0.5}{2 \cdot \log \alpha}}{\pi} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (/ (- (fma alpha alpha -0.5) 0.5) (* 2.0 (log alpha))) PI))

C

float code(float cosTheta, float alpha) {
	return ((fmaf(alpha, alpha, -0.5f) - 0.5f) / (2.0f * logf(alpha))) / ((float) M_PI);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(fma(alpha, alpha, Float32(-0.5)) - Float32(0.5)) / Float32(Float32(2.0) * log(alpha))) / Float32(pi))
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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.1%

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

8. Applied rewrites 95.2%

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

10. Applied rewrites 95.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -0.5\right) - 0.5}{2 \cdot \log \alpha}}{\pi} \]
11. Add Preprocessing

Alternative 10: 95.2% accurate, 1.8× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (/ (fma alpha alpha -1.0) (* 2.0 (log alpha))) PI))

C

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

Julia

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.1%

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

8. Applied rewrites 95.2%

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

Alternative 11: 95.2% accurate, 1.8× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (fma alpha alpha -1.0) (* (* PI 2.0) (log alpha))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.0%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 94.7%

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

8. Applied rewrites 95.1%

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

10. Applied rewrites 95.2%

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

Alternative 12: 95.0% accurate, 1.8× speedup

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

Math

\[\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{2 \cdot \log \alpha} \cdot 0.31830987334251404 \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (* (/ (fma alpha alpha -1.0) (* 2.0 (log alpha))) 0.31830987334251404))

C

float code(float cosTheta, float alpha) {
	return (fmaf(alpha, alpha, -1.0f) / (2.0f * logf(alpha))) * 0.31830987334251404f;
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(2.0) * log(alpha))) * Float32(0.31830987334251404))
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

3. Applied rewrites 98.2%

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

4. Evaluated real constant 98.2%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \log \left(\alpha \cdot \alpha\right)} \cdot 0.31830987334251404 \]
5. Step-by-step derivation

6. Applied rewrites 98.3%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right) \cdot \left(2 \cdot \log \alpha\right)} \cdot 0.31830987334251404 \]

7. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{2 \cdot \log \alpha} \cdot 0.31830987334251404 \]
8. Step-by-step derivation

9. Applied rewrites 95.0%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{2 \cdot \log \alpha} \cdot 0.31830987334251404 \]
10. Add Preprocessing

Alternative 13: 65.4% accurate, 1.9× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (/ 1.0 PI) (* -1.0 (* (log alpha) 2.0))))

C

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

Julia

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

MATLAB

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in alpha around 0

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

6. Applied rewrites 66.7%

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

7. Taylor expanded in cosTheta around 0

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

9. Applied rewrites 65.4%

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

11. Applied rewrites 65.4%

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

Alternative 14: 65.4% accurate, 2.4× speedup

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

Math

\[\frac{0.31830987334251404}{-1 \cdot \log \left(\alpha \cdot \alpha\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ 0.31830987334251404 (* -1.0 (log (* alpha alpha)))))

C

float code(float cosTheta, float alpha) {
	return 0.31830987334251404f / (-1.0f * logf((alpha * alpha)));
}

Fortran

real(4) function code(costheta, alpha)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: alpha
    code = 0.31830987334251404e0 / ((-1.0e0) * log((alpha * alpha)))
end function

Julia

function code(cosTheta, alpha)
	return Float32(Float32(0.31830987334251404) / Float32(Float32(-1.0) * log(Float32(alpha * alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.31830987334251404) / (single(-1.0) * log((alpha * alpha)));
end

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in alpha around 0

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

6. Applied rewrites 66.7%

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

7. Taylor expanded in cosTheta around 0

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

9. Applied rewrites 65.4%

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

10. Evaluated real constant 65.4%

    \[\leadsto \frac{0.31830987334251404}{-1 \cdot \log \left(\alpha \cdot \alpha\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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)))))