Average Error: 0.7 → 0.5
Time: 6.9s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{1 + \left(c + \frac{1}{\sqrt{\pi} \cdot cosTheta} \cdot \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]
(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
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (/ 1.0 (* (sqrt PI) cosTheta))
     (/ (sqrt (fma cosTheta -2.0 1.0)) (pow (exp cosTheta) cosTheta)))))))
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) {
	return 1.0f / (1.0f + (c + ((1.0f / (sqrtf(((float) M_PI)) * cosTheta)) * (sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / powf(expf(cosTheta), cosTheta)))));
}

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)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(1.0) / Float32(sqrt(Float32(pi)) * cosTheta)) * Float32(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / (exp(cosTheta) ^ cosTheta))))))
end

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

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

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

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2022150 
(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))))))