Average Error: 0.7 → 0.5
Time: 9.5s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \frac{1 + cosTheta \cdot -2}{\pi}\\ t_1 := 1 + \sqrt{t_0} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}\\ \left(\frac{1}{1 + \frac{\sqrt[3]{t_0}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{{\left(\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\right)}^{0.16666666666666666}}}} + \frac{{c}^{2}}{{t_1}^{3}}\right) - \left(\frac{{c}^{3}}{{t_1}^{4}} + \frac{c}{{t_1}^{2}}\right) \end{array} \]
(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 (/ (+ 1.0 (* cosTheta -2.0)) PI))
        (t_1
         (+ 1.0 (* (sqrt t_0) (/ 1.0 (* cosTheta (exp (pow cosTheta 2.0))))))))
   (-
    (+
     (/
      1.0
      (+
       1.0
       (/
        (cbrt t_0)
        (/
         (* cosTheta (pow (exp cosTheta) cosTheta))
         (pow (/ (fma cosTheta -2.0 1.0) PI) 0.16666666666666666)))))
     (/ (pow c 2.0) (pow t_1 3.0)))
    (+ (/ (pow c 3.0) (pow t_1 4.0)) (/ c (pow t_1 2.0))))))
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 = (1.0f + (cosTheta * -2.0f)) / ((float) M_PI);
	float t_1 = 1.0f + (sqrtf(t_0) * (1.0f / (cosTheta * expf(powf(cosTheta, 2.0f)))));
	return ((1.0f / (1.0f + (cbrtf(t_0) / ((cosTheta * powf(expf(cosTheta), cosTheta)) / powf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI)), 0.16666666666666666f))))) + (powf(c, 2.0f) / powf(t_1, 3.0f))) - ((powf(c, 3.0f) / powf(t_1, 4.0f)) + (c / powf(t_1, 2.0f)));
}

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)
	t_0 = Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / Float32(pi))
	t_1 = Float32(Float32(1.0) + Float32(sqrt(t_0) * Float32(Float32(1.0) / Float32(cosTheta * exp((cosTheta ^ Float32(2.0)))))))
	return Float32(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(cbrt(t_0) / Float32(Float32(cosTheta * (exp(cosTheta) ^ cosTheta)) / (Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi)) ^ Float32(0.16666666666666666)))))) + Float32((c ^ Float32(2.0)) / (t_1 ^ Float32(3.0)))) - Float32(Float32((c ^ Float32(3.0)) / (t_1 ^ Float32(4.0))) + Float32(c / (t_1 ^ Float32(2.0)))))
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}}
↓
\begin{array}{l}
t_0 := \frac{1 + cosTheta \cdot -2}{\pi}\\
t_1 := 1 + \sqrt{t_0} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}\\
\left(\frac{1}{1 + \frac{\sqrt[3]{t_0}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{{\left(\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\right)}^{0.16666666666666666}}}} + \frac{{c}^{2}}{{t_1}^{3}}\right) - \left(\frac{{c}^{3}}{{t_1}^{4}} + \frac{c}{{t_1}^{2}}\right)
\end{array}

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. Taylor expanded in c around 0 0.7

    \[\leadsto \color{blue}{\left(\frac{1}{1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}} + \frac{{c}^{2}}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{3}}\right) - \left(\frac{{c}^{3}}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{4}} + \frac{c}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{2}}\right)} \]

  4. Applied egg-rr0.5

    \[\leadsto \left(\frac{1}{1 + \color{blue}{\frac{\sqrt[3]{\frac{1 + cosTheta \cdot -2}{\pi}}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{\sqrt[3]{\sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}}}}}} + \frac{{c}^{2}}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{3}}\right) - \left(\frac{{c}^{3}}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{4}} + \frac{c}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{2}}\right) \]

  5. Applied egg-rr0.5

    \[\leadsto \left(\frac{1}{1 + \frac{\sqrt[3]{\frac{1 + cosTheta \cdot -2}{\pi}}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{\color{blue}{{\left(\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\right)}^{0.16666666666666666}}}}} + \frac{{c}^{2}}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{3}}\right) - \left(\frac{{c}^{3}}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{4}} + \frac{c}{{\left(1 + \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta}\right)}^{2}}\right) \]

  6. Final simplification0.5

    \[\leadsto \left(\frac{1}{1 + \frac{\sqrt[3]{\frac{1 + cosTheta \cdot -2}{\pi}}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{{\left(\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\right)}^{0.16666666666666666}}}} + \frac{{c}^{2}}{{\left(1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}\right)}^{3}}\right) - \left(\frac{{c}^{3}}{{\left(1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}\right)}^{4}} + \frac{c}{{\left(1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}}\right)}^{2}}\right) \]

Reproduce

herbie shell --seed 2022166 
(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))))))