Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 20.4s

Alternatives: 7

Speedup: 1.0×

Specification

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

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))));
}

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

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))));
}

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (/ (sqrt (fma cosTheta -2.0 1.0)) (* cosTheta (sqrt PI)))
    (exp (* cosTheta (- cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / (cosTheta * sqrtf(((float) M_PI)))) * expf((cosTheta * -cosTheta))));
}

Julia

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

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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 97.8%

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

   2. inv-pow 97.8%

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

   3. sqrt-pow2 97.8%

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

   4. metadata-eval 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. *-lft-identity 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Simplified 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
7. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{{\pi}^{-0.5} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate--l- 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{{\pi}^{-0.5} \cdot \sqrt{\color{blue}{1 - \left(cosTheta + cosTheta\right)}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

8. Applied egg-rr 98.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{{\pi}^{-0.5} \cdot \sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
9. Step-by-step derivation

   1. clear-num 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - \left(cosTheta + cosTheta\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. inv-pow 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\left(\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - \left(cosTheta + cosTheta\right)}}\right)}^{-1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. count-2 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + {\left(\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - \color{blue}{2 \cdot cosTheta}}}\right)}^{-1} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

10. Applied egg-rr 98.0%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\left(\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - 2 \cdot cosTheta}}\right)}^{-1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
11. Step-by-step derivation

    1. unpow-1 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - 2 \cdot cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    2. associate-/r* 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{1 - 2 \cdot cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    3. cancel-sign-sub-inv 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{1 + \left(-2\right) \cdot cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    4. metadata-eval 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{1 + \color{blue}{-2} \cdot cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    5. +-commutative 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{-2 \cdot cosTheta + 1}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    6. *-commutative 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{cosTheta \cdot -2} + 1}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    7. fma-undefine 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

12. Simplified 98.0%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
13. Step-by-step derivation

    1. associate-/r/ 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{1}{\frac{cosTheta}{{\pi}^{-0.5}}} \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    2. div-inv 98.8%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\color{blue}{cosTheta \cdot \frac{1}{{\pi}^{-0.5}}}} \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    3. pow-flip 98.8%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{cosTheta \cdot \color{blue}{{\pi}^{\left(--0.5\right)}}} \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    4. metadata-eval 98.8%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{cosTheta \cdot {\pi}^{\color{blue}{0.5}}} \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    5. pow1/2 98.8%

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

14. Applied egg-rr 98.8%

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

    1. associate-*l/ 98.9%

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

    2. *-lft-identity 98.9%

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

16. Simplified 98.9%

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

17. Final simplification 98.9%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (exp (* cosTheta (- cosTheta)))
    (/ (sqrt (/ (fma cosTheta -2.0 1.0) PI)) cosTheta)))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * (sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI))) / cosTheta)));
}

Julia

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

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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 97.8%

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

   2. inv-pow 97.8%

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

   3. sqrt-pow2 97.8%

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

   4. metadata-eval 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. *-lft-identity 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Simplified 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
7. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{{\pi}^{-0.5} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. associate--l- 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{{\pi}^{-0.5} \cdot \sqrt{\color{blue}{1 - \left(cosTheta + cosTheta\right)}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

8. Applied egg-rr 98.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{{\pi}^{-0.5} \cdot \sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
9. Step-by-step derivation

   1. clear-num 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - \left(cosTheta + cosTheta\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. inv-pow 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\left(\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - \left(cosTheta + cosTheta\right)}}\right)}^{-1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. count-2 98.0%

      \[\leadsto \frac{1}{\left(1 + c\right) + {\left(\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - \color{blue}{2 \cdot cosTheta}}}\right)}^{-1} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

10. Applied egg-rr 98.0%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\left(\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - 2 \cdot cosTheta}}\right)}^{-1}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
11. Step-by-step derivation

    1. unpow-1 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\frac{cosTheta}{{\pi}^{-0.5} \cdot \sqrt{1 - 2 \cdot cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    2. associate-/r* 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{1 - 2 \cdot cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    3. cancel-sign-sub-inv 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{1 + \left(-2\right) \cdot cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    4. metadata-eval 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{1 + \color{blue}{-2} \cdot cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    5. +-commutative 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{-2 \cdot cosTheta + 1}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    6. *-commutative 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{cosTheta \cdot -2} + 1}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    7. fma-undefine 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

12. Simplified 98.0%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\frac{\frac{cosTheta}{{\pi}^{-0.5}}}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
13. Step-by-step derivation

    1. associate-/l/ 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\frac{cosTheta}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    2. pow1/2 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{cosTheta}{\color{blue}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.5}} \cdot {\pi}^{-0.5}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    3. metadata-eval 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{cosTheta}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.5} \cdot {\pi}^{\color{blue}{\left(-1 \cdot 0.5\right)}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    4. pow-pow 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{cosTheta}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.5} \cdot \color{blue}{{\left({\pi}^{-1}\right)}^{0.5}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    5. inv-pow 98.0%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{cosTheta}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.5} \cdot {\color{blue}{\left(\frac{1}{\pi}\right)}}^{0.5}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    6. unpow-prod-down 98.2%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{cosTheta}{\color{blue}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right) \cdot \frac{1}{\pi}\right)}^{0.5}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    7. div-inv 98.2%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{cosTheta}{{\color{blue}{\left(\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\right)}}^{0.5}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    8. pow1/2 98.2%

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

    9. clear-num 98.2%

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

    10. *-un-lft-identity 98.2%

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

14. Applied egg-rr 98.2%

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

    1. *-lft-identity 98.2%

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

16. Simplified 98.2%

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

17. Final simplification 98.2%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{e^{-{cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - cosTheta \cdot 2}{\pi}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (/ (exp (- (pow cosTheta 2.0))) cosTheta)
    (sqrt (/ (- 1.0 (* cosTheta 2.0)) PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((expf(-powf(cosTheta, 2.0f)) / cosTheta) * sqrtf(((1.0f - (cosTheta * 2.0f)) / ((float) M_PI)))));
}

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((exp(-(cosTheta ^ single(2.0))) / cosTheta) * sqrt(((single(1.0) - (cosTheta * single(2.0))) / single(pi)))));
end

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. Add Preprocessing

3. Taylor expanded in c around 0 97.5%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]

4. Final simplification 97.5%

   \[\leadsto \frac{1}{1 + \frac{e^{-{cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - cosTheta \cdot 2}{\pi}}} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \left(\sqrt{\pi} + \left(cosTheta \cdot \pi\right) \cdot \left({\pi}^{-0.5} + \left(-1 - c\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (* cosTheta (+ (sqrt PI) (* (* cosTheta PI) (+ (pow PI -0.5) (- -1.0 c))))))

C

float code(float cosTheta, float c) {
	return cosTheta * (sqrtf(((float) M_PI)) + ((cosTheta * ((float) M_PI)) * (powf(((float) M_PI), -0.5f) + (-1.0f - c))));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(Float32(cosTheta * Float32(pi)) * Float32((Float32(pi) ^ Float32(-0.5)) + Float32(Float32(-1.0) - c)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + ((cosTheta * single(pi)) * ((single(pi) ^ single(-0.5)) + (single(-1.0) - c))));
end

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. Step-by-step derivation

   1. associate-+l+ 97.8%

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

   2. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate--l- 98.9%

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

   7. sub-neg 98.9%

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

   8. neg-mul-1 98.9%

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

   9. distribute-lft-out 98.9%

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

   10. distribute-rgt-out 98.9%

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

   11. metadata-eval 98.9%

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

   12. exp-prod 98.9%

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

3. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. times-frac 97.8%

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

   3. inv-pow 97.8%

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

   4. sqrt-pow2 97.8%

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

   5. metadata-eval 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   6. +-commutative 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5} \cdot \frac{\sqrt{\color{blue}{cosTheta \cdot -2 + 1}}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   7. fma-define 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5} \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

6. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{{\pi}^{-0.5} \cdot \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]
7. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta} \cdot {\pi}^{-0.5}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   2. associate-*l/ 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   3. *-lft-identity 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}{\color{blue}{1 \cdot cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   4. times-frac 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{1} \cdot \frac{{\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   5. /-rgt-identity 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot \frac{{\pi}^{-0.5}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{{\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

9. Taylor expanded in cosTheta around 0 95.9%

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

    1. mul-1-neg 95.9%

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

    2. unsub-neg 95.9%

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

    3. associate-*r* 95.9%

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

    4. mul-1-neg 95.9%

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

    5. unsub-neg 95.9%

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

11. Simplified 95.9%

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

    1. associate-+r- 95.9%

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

    2. pow1/2 95.9%

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(\left(1 + c\right) - \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)\right) \]

    3. inv-pow 95.9%

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(\left(1 + c\right) - {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)\right) \]

    4. pow-pow 95.9%

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(\left(1 + c\right) - \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)\right) \]

    5. metadata-eval 95.9%

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(\left(1 + c\right) - {\pi}^{\color{blue}{-0.5}}\right)\right) \]

13. Applied egg-rr 95.9%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \color{blue}{\left(\left(1 + c\right) - {\pi}^{-0.5}\right)}\right) \]

14. Final simplification 95.9%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + \left(cosTheta \cdot \pi\right) \cdot \left({\pi}^{-0.5} + \left(-1 - c\right)\right)\right) \]
15. Add Preprocessing

Alternative 5: 95.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(1 - \sqrt{\frac{1}{\pi}}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (* cosTheta (- (sqrt PI) (* (* cosTheta PI) (- 1.0 (sqrt (/ 1.0 PI)))))))

C

float code(float cosTheta, float c) {
	return cosTheta * (sqrtf(((float) M_PI)) - ((cosTheta * ((float) M_PI)) * (1.0f - sqrtf((1.0f / ((float) M_PI))))));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) - Float32(Float32(cosTheta * Float32(pi)) * Float32(Float32(1.0) - sqrt(Float32(Float32(1.0) / Float32(pi)))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) - ((cosTheta * single(pi)) * (single(1.0) - sqrt((single(1.0) / single(pi))))));
end

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. Step-by-step derivation

   1. associate-+l+ 97.8%

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

   2. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate--l- 98.9%

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

   7. sub-neg 98.9%

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

   8. neg-mul-1 98.9%

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

   9. distribute-lft-out 98.9%

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

   10. distribute-rgt-out 98.9%

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

   11. metadata-eval 98.9%

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

   12. exp-prod 98.9%

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

3. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. times-frac 97.8%

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

   3. inv-pow 97.8%

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

   4. sqrt-pow2 97.8%

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

   5. metadata-eval 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   6. +-commutative 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5} \cdot \frac{\sqrt{\color{blue}{cosTheta \cdot -2 + 1}}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   7. fma-define 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5} \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

6. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{{\pi}^{-0.5} \cdot \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]
7. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta} \cdot {\pi}^{-0.5}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   2. associate-*l/ 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   3. *-lft-identity 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}{\color{blue}{1 \cdot cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   4. times-frac 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{1} \cdot \frac{{\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   5. /-rgt-identity 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot \frac{{\pi}^{-0.5}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{{\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

9. Taylor expanded in cosTheta around 0 95.9%

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

    1. mul-1-neg 95.9%

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

    2. unsub-neg 95.9%

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

    3. associate-*r* 95.9%

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

    4. mul-1-neg 95.9%

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

    5. unsub-neg 95.9%

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

11. Simplified 95.9%

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

12. Taylor expanded in c around 0 95.6%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \color{blue}{\left(1 - \sqrt{\frac{1}{\pi}}\right)}\right) \]

13. Final simplification 95.6%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - \left(cosTheta \cdot \pi\right) \cdot \left(1 - \sqrt{\frac{1}{\pi}}\right)\right) \]
14. Add Preprocessing

Alternative 6: 93.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \sqrt{\pi} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (* cosTheta (sqrt PI)))

C

float code(float cosTheta, float c) {
	return cosTheta * sqrtf(((float) M_PI));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * sqrt(Float32(pi)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * sqrt(single(pi));
end

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. Step-by-step derivation

   1. associate-+l+ 97.8%

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

   2. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate--l- 98.9%

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

   7. sub-neg 98.9%

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

   8. neg-mul-1 98.9%

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

   9. distribute-lft-out 98.9%

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

   10. distribute-rgt-out 98.9%

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

   11. metadata-eval 98.9%

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

   12. exp-prod 98.9%

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

3. Simplified 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. times-frac 97.8%

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

   3. inv-pow 97.8%

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

   4. sqrt-pow2 97.8%

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

   5. metadata-eval 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{\color{blue}{-0.5}} \cdot \frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   6. +-commutative 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5} \cdot \frac{\sqrt{\color{blue}{cosTheta \cdot -2 + 1}}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   7. fma-define 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5} \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

6. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{{\pi}^{-0.5} \cdot \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]
7. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta} \cdot {\pi}^{-0.5}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   2. associate-*l/ 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   3. *-lft-identity 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot {\pi}^{-0.5}}{\color{blue}{1 \cdot cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   4. times-frac 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{1} \cdot \frac{{\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

   5. /-rgt-identity 98.1%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot \frac{{\pi}^{-0.5}}{cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

8. Simplified 98.1%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{{\pi}^{-0.5}}{cosTheta}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

9. Taylor expanded in cosTheta around 0 93.1%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]

10. Final simplification 93.1%

    \[\leadsto cosTheta \cdot \sqrt{\pi} \]
11. Add Preprocessing

Alternative 7: 5.1% accurate, 107.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (/ 1.0 c))

C

float code(float cosTheta, float c) {
	return 1.0f / c;
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 / c
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / c)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / c;
end

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. Step-by-step derivation

   1. associate-+l+ 97.8%

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

   2. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate--l- 98.9%

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

   7. sub-neg 98.9%

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

   8. neg-mul-1 98.9%

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

   9. distribute-lft-out 98.9%

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

   10. distribute-rgt-out 98.9%

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

   11. metadata-eval 98.9%

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

   12. exp-prod 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in c around inf 4.7%

   \[\leadsto \color{blue}{\frac{1}{c}} \]

6. Final simplification 4.7%

   \[\leadsto \frac{1}{c} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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))))))