GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 5.6s

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: 98.5% 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

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

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 (* alpha alpha))
  (* PI (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0)))))

C

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

Julia

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

3. Applied rewrites 98.5%

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

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

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)
 (*
  PI
  (*
   (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0)
   (log (* alpha alpha))))))

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(pi) * Float32(fma(Float32(cosTheta * cosTheta), fma(alpha, alpha, Float32(-1.0)), Float32(1.0)) * 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 98.5%

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

Alternative 3: 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

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

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)
 (*
  (fma (* cosTheta cosTheta) (fma alpha alpha -1.0) 1.0)
  (* (* 2.0 (log 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) * ((2.0f * logf(alpha)) * ((float) M_PI)));
}

Julia

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

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

4. Taylor expanded in alpha around 0

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

6. Applied rewrites 98.5%

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

Alternative 4: 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

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (fma 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 fmaf(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(fma(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 98.4%

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

5. Applied rewrites 98.4%

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

Alternative 5: 97.5% accurate, 1.2× speedup

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

Alternative 6: 97.5% 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

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

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)
 (* PI (* (log (* alpha alpha)) (fma (- cosTheta) cosTheta 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 97.5%

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

Alternative 7: 97.5% 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

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

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)
 (* (fma (- cosTheta) cosTheta 1.0) (* (* 2.0 (log alpha)) PI))))

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(fma(Float32(-cosTheta), cosTheta, Float32(1.0)) * Float32(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. 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 97.4%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 97.5%

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

Alternative 8: 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

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

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)
 (* PI (* (fma (- cosTheta) cosTheta 1.0) (log (* alpha alpha))))))

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(pi) * Float32(fma(Float32(-cosTheta), cosTheta, Float32(1.0)) * 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. 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 97.4%

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

8. Applied rewrites 97.4%

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

Alternative 9: 95.4% 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

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

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) (* PI (log (* alpha alpha)))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / (single(pi) * 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 98.7%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 95.3%

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

Alternative 10: 95.3% 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

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.3%

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

8. Applied rewrites 95.4%

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

Alternative 11: 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 66.5%

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

6. Applied rewrites 66.5%

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

8. Applied rewrites 66.6%

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

9. Taylor expanded in cosTheta around 0

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

11. Applied rewrites 65.4%

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

Alternative 12: 65.3% 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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = 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. Step-by-step derivation

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.3%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 65.3%

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

11. Applied rewrites 65.3%

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

Alternative 13: 65.3% 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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-1.0) / (single(pi) * 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 98.7%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.5%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 65.4%

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

11. Applied rewrites 65.3%

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

Reproduce

herbie shell --seed 2026086 
(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)))))