Beckmann Sample, normalization factor

?

Percentage Accurate: 97.8% → 98.5%
Time: 11.6s
Precision: binary32
Cost: 10208

?

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{cosTheta \cdot \sqrt{\frac{\pi}{1 - \left(cosTheta + cosTheta\right)}}}} \end{array} \]
(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
 (/
  1.0
  (+
   (+ 1.0 c)
   (/
    (exp (* cosTheta (- cosTheta)))
    (* cosTheta (sqrt (/ PI (- 1.0 (+ 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) + (expf((cosTheta * -cosTheta)) / (cosTheta * sqrtf((((float) M_PI) / (1.0f - (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(Float32(1.0) + c) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) / Float32(cosTheta * sqrt(Float32(Float32(pi) / Float32(Float32(1.0) - Float32(cosTheta + cosTheta))))))))
end

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 = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp((cosTheta * -cosTheta)) / (cosTheta * sqrt((single(pi) / (single(1.0) - (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}}
↓
\begin{array}{l}

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

Local Percentage Accuracy vs ?

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.

Herbie found 10 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 97.8%

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

  2. Applied egg-rr98.5%

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

    [Start]97.8%

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

    frac-times [=>]98.5%

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

    *-un-lft-identity [<=]98.5%

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

    associate--r+ [<=]98.5%

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

  3. Applied egg-rr97.9%

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

    [Start]98.5%

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

    add-sqr-sqrt [=>]97.9%

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

    *-commutative [=>]97.9%

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

    *-commutative [=>]97.9%

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

    exp-prod [=>]98.0%

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

  4. Applied egg-rr98.5%

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

    [Start]97.9%

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

    add-sqr-sqrt [<=]98.5%

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

    clear-num [=>]98.5%

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

    *-commutative [=>]98.5%

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

    associate-*l/ [<=]98.5%

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

    add-sqr-sqrt [=>]97.8%

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

    pow-exp [=>]97.8%

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

    *-commutative [<=]97.8%

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

  5. Final simplification98.5%

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

Alternatives

Alternative 1
Accuracy98.5%
Cost10208
\[\frac{1}{\left(1 + c\right) + \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{cosTheta \cdot \sqrt{\frac{\pi}{1 - \left(cosTheta + cosTheta\right)}}}} \]
Alternative 2
Accuracy97.5%
Cost10144
\[\frac{1}{1 + \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{cosTheta} \cdot \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}} \]
Alternative 3
Accuracy96.5%
Cost6976
\[\frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(cosTheta \cdot -1.5 + \frac{1}{cosTheta}\right) + -1\right)} \]
Alternative 4
Accuracy95.3%
Cost6848
\[\frac{1}{1 + \left(c + \left(\frac{1}{cosTheta} + -1\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \]
Alternative 5
Accuracy95.6%
Cost6848
\[\frac{1}{1 + \left(\frac{1}{cosTheta} + \left(-1 - cosTheta\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]
Alternative 6
Accuracy96.2%
Cost6848
\[\frac{1}{\left(1 + c\right) + \frac{\left(\frac{1}{cosTheta} + -1\right) - cosTheta}{\sqrt{\pi}}} \]
Alternative 7
Accuracy95.2%
Cost6784
\[\frac{1}{1 + \left(\frac{1}{cosTheta} + -1\right) \cdot \sqrt{\frac{1}{\pi}}} \]
Alternative 8
Accuracy93.5%
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 9
Accuracy10.6%
Cost96
\[1 - c \]
Alternative 10
Accuracy10.6%
Cost32
\[1 \]

Reproduce?

herbie shell --seed 2023167 
(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))))))