Beckmann Sample, normalization factor

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Percentage Accurate: 97.8% → 98.4%
Time: 17.4s
Precision: binary32
Cost: 19712

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\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{\mathsf{fma}\left(\frac{\frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta}}{\sqrt{\pi}}, {\left(e^{cosTheta}\right)}^{\left(-cosTheta\right)}, 1 + c\right)} \]
(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
  (fma
   (/ (/ (sqrt (- 1.0 (+ cosTheta cosTheta))) cosTheta) (sqrt PI))
   (pow (exp cosTheta) (- cosTheta))
   (+ 1.0 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) {
	return 1.0f / fmaf(((sqrtf((1.0f - (cosTheta + cosTheta))) / cosTheta) / sqrtf(((float) M_PI))), powf(expf(cosTheta), -cosTheta), (1.0f + c));
}

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) / fma(Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(cosTheta + cosTheta))) / cosTheta) / sqrt(Float32(pi))), (exp(cosTheta) ^ Float32(-cosTheta)), Float32(Float32(1.0) + c)))
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}{\mathsf{fma}\left(\frac{\frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta}}{\sqrt{\pi}}, {\left(e^{cosTheta}\right)}^{\left(-cosTheta\right)}, 1 + c\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.

Herbie found 12 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

Derivation?

  1. Initial program 98.0%

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

  2. Simplified98.6%

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

    [Start]98.0

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

    +-commutative [=>]98.0

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

    fma-def [=>]98.0

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

    associate-*l/ [=>]98.6

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

    *-lft-identity [=>]98.6

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

    associate--l- [=>]98.6

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

    *-commutative [=>]98.6

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

    exp-prod [=>]98.6

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

  3. Final simplification98.6%

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

Alternatives

Alternative 1
Accuracy98.5%
Cost19680
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)} \]
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
Accuracy98.1%
Cost13344
\[\frac{1}{1 + \left(c + \frac{e^{-cosTheta \cdot cosTheta} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} \]
Alternative 4
Accuracy97.9%
Cost13344
\[\frac{1}{1 + \mathsf{fma}\left(\sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}, \frac{e^{-cosTheta \cdot cosTheta}}{cosTheta}, c\right)} \]
Alternative 5
Accuracy97.5%
Cost10400
\[\frac{1}{1 + \frac{e^{-cosTheta \cdot cosTheta}}{cosTheta} \cdot \sqrt{\frac{1 + \left(cosTheta \cdot cosTheta\right) \cdot -4}{\pi \cdot \left(1 + cosTheta \cdot 2\right)}}} \]
Alternative 6
Accuracy96.9%
Cost10176
\[\frac{1}{1 + \left(c + \left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1.5, 1\right) - cosTheta\right) \cdot \frac{1}{cosTheta \cdot \sqrt{\pi}}\right)} \]
Alternative 7
Accuracy96.3%
Cost6976
\[\frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1}{cosTheta} + -1\right) + cosTheta \cdot -1.5\right)\right)} \]
Alternative 8
Accuracy95.5%
Cost6848
\[\frac{1}{1 + \left(c + \frac{1}{cosTheta \cdot \sqrt{\pi}} \cdot \left(1 - cosTheta\right)\right)} \]
Alternative 9
Accuracy92.9%
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 10
Accuracy10.8%
Cost96
\[1 - c \]
Alternative 11
Accuracy10.8%
Cost32
\[1 \]

Reproduce?

herbie shell --seed 2023161 
(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))))))