Beckmann Sample, normalization factor

?

Percentage Accurate: 97.8% → 98.5%
Time: 18.0s
Precision: binary32
Cost: 19680

?

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]
(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
  (+
   c
   (+
    1.0
    (/
     (/ (sqrt (fma cosTheta -2.0 1.0)) (* cosTheta (sqrt PI)))
     (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 / (c + (1.0f + ((sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / (cosTheta * sqrtf(((float) M_PI)))) / 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(c + Float32(Float32(1.0) + Float32(Float32(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / Float32(cosTheta * sqrt(Float32(pi)))) / (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}{c + \left(1 + \frac{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Derivation?

  1. Initial program 97.7%

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

  2. Simplified98.4%

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

    [Start]97.7

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

    +-commutative [=>]97.7

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

    associate-+l+ [=>]97.7

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

    distribute-lft-neg-out [=>]97.7

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

    exp-neg [=>]97.7

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

    associate-*r/ [=>]97.7

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

    associate-/l* [=>]97.7

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

    /-rgt-identity [=>]97.7

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

  3. Final simplification98.4%

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

Alternatives

Alternative 1
Accuracy98.3%
Cost19680
\[\frac{1}{1 + \left(c + \frac{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}}{\sqrt{\pi}}\right)} \]
Alternative 2
Accuracy98.4%
Cost16544
\[\frac{1}{\left(1 + c\right) + \frac{\frac{\frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta}}{\sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \]
Alternative 3
Accuracy98.0%
Cost13312
\[\frac{1}{c + \left(1 + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 4
Accuracy97.6%
Cost10272
\[\frac{1}{1 + \left(c + \frac{\left(cosTheta \cdot -1.5 + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) + -1}{\sqrt{\pi}}\right)} \]
Alternative 5
Accuracy97.6%
Cost10176
\[\frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}} \]
Alternative 6
Accuracy97.0%
Cost6912
\[\frac{1}{1 + \left(c + \frac{\left(cosTheta \cdot -1.5 + \frac{1}{cosTheta}\right) + -1}{\sqrt{\pi}}\right)} \]
Alternative 7
Accuracy95.0%
Cost6784
\[\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + -1\right)} \]
Alternative 8
Accuracy95.7%
Cost6784
\[\frac{1}{1 + \left(c + \frac{\frac{1}{cosTheta} + -1}{\sqrt{\pi}}\right)} \]
Alternative 9
Accuracy93.1%
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 10
Accuracy10.9%
Cost3424
\[\pi \cdot \left(\left(cosTheta \cdot cosTheta\right) \cdot \left(-c\right)\right) \]
Alternative 11
Accuracy10.9%
Cost3424
\[\left(cosTheta \cdot cosTheta\right) \cdot \left(\pi \cdot \left(-c\right)\right) \]
Alternative 12
Accuracy10.7%
Cost96
\[1 - c \]
Alternative 13
Accuracy10.7%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023159 
(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))))))