Average Error: 0.7 → 0.4
Time: 13.2s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)\\ t_1 := \mathsf{fma}\left(c, c + -1, 1\right)\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t_0 \cdot t_1}{\mathsf{fma}\left(t_0, 1 + {c}^{3}, t_1 \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}\right)\right) \end{array} \]
\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}}
↓
\begin{array}{l}
t_0 := \sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)\\
t_1 := \mathsf{fma}\left(c, c + -1, 1\right)\\
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t_0 \cdot t_1}{\mathsf{fma}\left(t_0, 1 + {c}^{3}, t_1 \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}\right)\right)
\end{array}

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))
↓
(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (* (sqrt PI) (* cosTheta (exp (* cosTheta cosTheta)))))
        (t_1 (fma c (+ c -1.0) 1.0)))
   (expm1
    (log1p
     (/
      (* t_0 t_1)
      (fma t_0 (+ 1.0 (pow c 3.0)) (* t_1 (sqrt (fma cosTheta -2.0 1.0)))))))))
float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf((float) M_PI)) * (sqrtf((1.0f - cosTheta) - cosTheta) / cosTheta)) * expf(-cosTheta * cosTheta)));
}
↓
float code(float cosTheta, float c) {
	float t_0 = sqrtf((float) M_PI) * (cosTheta * expf(cosTheta * cosTheta));
	float t_1 = fmaf(c, (c + -1.0f), 1.0f);
	return expm1f(log1pf((t_0 * t_1) / fmaf(t_0, (1.0f + powf(c, 3.0f)), (t_1 * sqrtf(fmaf(cosTheta, -2.0f, 1.0f))))));
}

Error

Bits error versus cosTheta

Bits error versus c

Derivation

  1. Initial program 0.7

    \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{1}{\left(1 + c\right) + \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)}}} \]

  3. Applied flip3-+_binary320.5

    \[\leadsto \frac{1}{\color{blue}{\frac{{1}^{3} + {c}^{3}}{1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)}} + \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)}} \]

  4. Applied frac-add_binary320.5

    \[\leadsto \frac{1}{\color{blue}{\frac{\left({1}^{3} + {c}^{3}\right) \cdot \left(\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right) + \left(1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\left(1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)}}} \]

  5. Applied associate-/r/_binary320.4

    \[\leadsto \color{blue}{\frac{1}{\left({1}^{3} + {c}^{3}\right) \cdot \left(\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right) + \left(1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot \left(\left(1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)\right)} \]

  6. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right), 1 + {c}^{3}, \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \mathsf{fma}\left(c, c + -1, 1\right)\right)}} \cdot \left(\left(1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)\right) \]

  7. Applied expm1-log1p-u_binary320.4

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right), 1 + {c}^{3}, \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \mathsf{fma}\left(c, c + -1, 1\right)\right)} \cdot \left(\left(1 \cdot 1 + \left(c \cdot c - 1 \cdot c\right)\right) \cdot \left(\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)\right)\right)\right)} \]

  8. Simplified0.4

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)\right) \cdot \mathsf{fma}\left(c, c + -1, 1\right)}{\mathsf{fma}\left(\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right), 1 + {c}^{3}, \mathsf{fma}\left(c, c + -1, 1\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}\right)}\right) \]

  9. Final simplification0.4

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)\right) \cdot \mathsf{fma}\left(c, c + -1, 1\right)}{\mathsf{fma}\left(\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right), 1 + {c}^{3}, \mathsf{fma}\left(c, c + -1, 1\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022082 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))