Beckmann Sample, normalization factor

Average Error: 0.7 → 0.4

Time: 17.1s

Precision: binary32

Cost: 10340

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;cosTheta \leq 2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;cosTheta \cdot \sqrt{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c + \left(1 + \frac{\sqrt{\frac{-1 + cosTheta \cdot 2}{cosTheta \cdot \left(\left(-cosTheta\right) \cdot \pi\right)}}}{e^{cosTheta \cdot cosTheta}}\right)}\\ \end{array} \]

FPCore

(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
 (if (<= cosTheta 2.4999999292951713e-10)
   (* cosTheta (sqrt PI))
   (/
    1.0
    (+
     c
     (+
      1.0
      (/
       (sqrt (/ (+ -1.0 (* cosTheta 2.0)) (* cosTheta (* (- cosTheta) PI))))
       (exp (* cosTheta cosTheta))))))))

C

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 tmp;
	if (cosTheta <= 2.4999999292951713e-10f) {
		tmp = cosTheta * sqrtf(((float) M_PI));
	} else {
		tmp = 1.0f / (c + (1.0f + (sqrtf(((-1.0f + (cosTheta * 2.0f)) / (cosTheta * (-cosTheta * ((float) M_PI))))) / expf((cosTheta * cosTheta)))));
	}
	return tmp;
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

↓

function code(cosTheta, c)
	tmp = Float32(0.0)
	if (cosTheta <= Float32(2.4999999292951713e-10))
		tmp = Float32(cosTheta * sqrt(Float32(pi)));
	else
		tmp = Float32(Float32(1.0) / Float32(c + Float32(Float32(1.0) + Float32(sqrt(Float32(Float32(Float32(-1.0) + Float32(cosTheta * Float32(2.0))) / Float32(cosTheta * Float32(Float32(-cosTheta) * Float32(pi))))) / exp(Float32(cosTheta * cosTheta))))));
	end
	return tmp
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

↓

function tmp_2 = code(cosTheta, c)
	tmp = single(0.0);
	if (cosTheta <= single(2.4999999292951713e-10))
		tmp = cosTheta * sqrt(single(pi));
	else
		tmp = single(1.0) / (c + (single(1.0) + (sqrt(((single(-1.0) + (cosTheta * single(2.0))) / (cosTheta * (-cosTheta * single(pi))))) / exp((cosTheta * cosTheta)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta < 2.49999993e-10

   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. Simplified 0.5

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

      [Start] 0.7	\[ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
      +-commutative [=>] 0.7	\[ \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
      associate-+l+ [=>] 0.7	\[ \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]
      distribute-lft-neg-out [=>] 0.7	\[ \frac{1}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\color{blue}{-cosTheta \cdot cosTheta}}\right)} \]
      exp-neg [=>] 0.7	\[ \frac{1}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \color{blue}{\frac{1}{e^{cosTheta \cdot cosTheta}}}\right)} \]
      associate-*r/ [=>] 0.7	\[ \frac{1}{c + \left(1 + \color{blue}{\frac{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot 1}{e^{cosTheta \cdot cosTheta}}}\right)} \]
      associate-/l* [=>] 0.7	\[ \frac{1}{c + \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}}\right)} \]
      associate-*l/ [=>] 0.5	\[ \frac{1}{c + \left(1 + \frac{\color{blue}{\frac{1 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{\pi}}}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}\right)} \]
      *-lft-identity [=>] 0.5	\[ \frac{1}{c + \left(1 + \frac{\frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt{\pi}}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}\right)} \]

   3. Taylor expanded in cosTheta around 0 0.4

      \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]

   if 2.49999993e-10 < cosTheta

   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. Simplified 0.6

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

      [Start] 0.7	\[ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
      +-commutative [=>] 0.7	\[ \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
      associate-+l+ [=>] 0.7	\[ \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]
      distribute-lft-neg-out [=>] 0.7	\[ \frac{1}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\color{blue}{-cosTheta \cdot cosTheta}}\right)} \]
      exp-neg [=>] 0.7	\[ \frac{1}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \color{blue}{\frac{1}{e^{cosTheta \cdot cosTheta}}}\right)} \]
      associate-*r/ [=>] 0.7	\[ \frac{1}{c + \left(1 + \color{blue}{\frac{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot 1}{e^{cosTheta \cdot cosTheta}}}\right)} \]
      associate-/l* [=>] 0.7	\[ \frac{1}{c + \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}}\right)} \]
      associate-*l/ [=>] 0.6	\[ \frac{1}{c + \left(1 + \frac{\color{blue}{\frac{1 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{\pi}}}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}\right)} \]
      *-lft-identity [=>] 0.6	\[ \frac{1}{c + \left(1 + \frac{\frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt{\pi}}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}\right)} \]

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.5

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

   5. Simplified 0.4

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

      [Start] 0.5	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\left(-1 - cosTheta \cdot -2\right) \cdot \frac{1}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      *-commutative [<=] 0.5	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\color{blue}{\frac{1}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)} \cdot \left(-1 - cosTheta \cdot -2\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      associate-*l/ [=>] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\color{blue}{\frac{1 \cdot \left(-1 - cosTheta \cdot -2\right)}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      *-lft-identity [=>] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\frac{\color{blue}{-1 - cosTheta \cdot -2}}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      cancel-sign-sub-inv [=>] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\frac{\color{blue}{-1 + \left(-cosTheta\right) \cdot -2}}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      distribute-lft-neg-in [<=] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\frac{-1 + \color{blue}{\left(-cosTheta \cdot -2\right)}}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      distribute-rgt-neg-in [=>] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\frac{-1 + \color{blue}{cosTheta \cdot \left(--2\right)}}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      metadata-eval [=>] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\frac{-1 + cosTheta \cdot \color{blue}{2}}{\left(cosTheta \cdot \pi\right) \cdot \left(-cosTheta\right)}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
      associate-*l* [=>] 0.4	\[ \frac{1}{c + \left(1 + \frac{\sqrt{\frac{-1 + cosTheta \cdot 2}{\color{blue}{cosTheta \cdot \left(\pi \cdot \left(-cosTheta\right)\right)}}}}{e^{cosTheta \cdot cosTheta}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta \leq 2.4999999292951713 \cdot 10^{-10}:\\ \;\;\;\;cosTheta \cdot \sqrt{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c + \left(1 + \frac{\sqrt{\frac{-1 + cosTheta \cdot 2}{cosTheta \cdot \left(\left(-cosTheta\right) \cdot \pi\right)}}}{e^{cosTheta \cdot cosTheta}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	22848

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

Alternative 2
Error	0.5
Cost	13376

\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]

Alternative 3
Error	0.6
Cost	10272

\[\frac{1}{\left(1 + c\right) + \frac{\frac{1}{\sqrt{\frac{-\pi}{-1 + cosTheta \cdot 2}}}}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}} \]

Alternative 4
Error	1.2
Cost	6976

\[\frac{1}{\left(1 + c\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\left(-1 + \frac{1}{cosTheta}\right) + cosTheta \cdot -1.5\right)} \]

Alternative 5
Error	1.7
Cost	6848

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

Alternative 6
Error	2.3
Cost	6464

\[cosTheta \cdot \sqrt{\pi} \]

Alternative 7
Error	28.5
Cost	96

\[1 - c \]

Alternative 8
Error	28.5
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023027 
(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))))))