Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.6%

Time: 21.2s

Alternatives: 10

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: 97.9% 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: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. fma-define 97.6%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{{\pi}^{-0.16666666666666666}}{\sqrt[3]{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (*
     (/ (pow PI -0.16666666666666666) (cbrt PI))
     (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* cosTheta (- cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((powf(((float) M_PI), -0.16666666666666666f) / cbrtf(((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(pi) ^ Float32(-0.16666666666666666)) / cbrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

Derivation

1. Initial program 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 97.6%

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

   2. inv-pow 97.6%

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

   3. sqrt-pow2 97.6%

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

   4. metadata-eval 97.6%

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

4. Applied egg-rr 97.6%

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

   1. *-lft-identity 97.6%

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

6. Simplified 97.6%

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

   1. add-cbrt-cube 97.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\sqrt[3]{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right) \cdot {\pi}^{-0.5}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. pow-prod-up 97.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\sqrt[3]{\color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}} \cdot {\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. metadata-eval 97.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\sqrt[3]{{\pi}^{\color{blue}{-1}} \cdot {\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. inv-pow 97.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\sqrt[3]{\color{blue}{\frac{1}{\pi}} \cdot {\pi}^{-0.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. cbrt-unprod 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. *-commutative 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\sqrt[3]{{\pi}^{-0.5}} \cdot \sqrt[3]{\frac{1}{\pi}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   7. pow1/3 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\color{blue}{{\left({\pi}^{-0.5}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\frac{1}{\pi}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. pow-pow 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\color{blue}{{\pi}^{\left(-0.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt[3]{\frac{1}{\pi}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   9. metadata-eval 97.8%

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

   10. cbrt-div 97.8%

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

   11. metadata-eval 97.8%

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

8. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left({\pi}^{-0.16666666666666666} \cdot \frac{1}{\sqrt[3]{\pi}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
9. Step-by-step derivation

   1. associate-*r/ 97.8%

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

   2. *-rgt-identity 97.8%

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

10. Simplified 97.8%

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

11. Final simplification 97.8%

    \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{{\pi}^{-0.16666666666666666}}{\sqrt[3]{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \]
12. Add Preprocessing

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 97.6%

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

   2. inv-pow 97.6%

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

   3. sqrt-pow2 97.6%

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

   4. metadata-eval 97.6%

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

4. Applied egg-rr 97.6%

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

   1. *-lft-identity 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{{\pi}^{-0.5} \cdot \sqrt{1 - cosTheta \cdot 2}}{cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (exp (* cosTheta (- cosTheta)))
    (/ (* (pow PI -0.5) (sqrt (- 1.0 (* cosTheta 2.0)))) cosTheta)))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * ((powf(((float) M_PI), -0.5f) * sqrtf((1.0f - (cosTheta * 2.0f)))) / cosTheta)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) * Float32(Float32((Float32(pi) ^ Float32(-0.5)) * sqrt(Float32(Float32(1.0) - Float32(cosTheta * Float32(2.0))))) / cosTheta))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp((cosTheta * -cosTheta)) * (((single(pi) ^ single(-0.5)) * sqrt((single(1.0) - (cosTheta * single(2.0))))) / cosTheta)));
end

Derivation

1. Initial program 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 97.6%

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

   2. cbrt-unprod 97.6%

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

   3. frac-times 97.8%

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

   4. metadata-eval 97.8%

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

   5. add-sqr-sqrt 97.8%

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

   6. inv-pow 97.8%

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

   7. sqrt-pow2 97.8%

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

   8. metadata-eval 97.8%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{\color{blue}{-0.5}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. cbrt-unprod 97.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\sqrt[3]{\frac{1}{\pi} \cdot {\pi}^{-0.5}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. add-cbrt-cube 45.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\sqrt[3]{\frac{1}{\pi} \cdot {\pi}^{-0.5}} \cdot \color{blue}{\sqrt[3]{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. cbrt-unprod 45.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\sqrt[3]{\left(\frac{1}{\pi} \cdot {\pi}^{-0.5}\right) \cdot \left(\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. inv-pow 45.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt[3]{\left(\color{blue}{{\pi}^{-1}} \cdot {\pi}^{-0.5}\right) \cdot \left(\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. metadata-eval 45.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt[3]{\left({\pi}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot {\pi}^{-0.5}\right) \cdot \left(\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. pow-prod-up 45.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt[3]{\left(\color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot {\pi}^{-0.5}\right) \cdot \left(\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   7. cbrt-unprod 45.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\sqrt[3]{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right) \cdot {\pi}^{-0.5}} \cdot \sqrt[3]{\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied egg-rr 97.7%

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

7. Final simplification 97.7%

   \[\leadsto \frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{{\pi}^{-0.5} \cdot \sqrt{1 - cosTheta \cdot 2}}{cosTheta}} \]
8. Add Preprocessing

Alternative 5: 97.7% 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 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. fma-define 97.6%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \cdot \sqrt{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}}} \]

   2. pow2 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}}\right)}^{2}} \]

8. Applied egg-rr 97.8%

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

9. Taylor expanded in c around 0 97.4%

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

10. Final simplification 97.4%

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

Alternative 6: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. fma-define 97.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 95.9%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right)} \]
6. 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. *-commutative 95.9%

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

   4. associate-*l* 95.9%

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

   5. mul-1-neg 95.9%

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

   6. unsub-neg 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 7: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

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))) - c))));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. fma-define 97.6%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 97.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \cdot \sqrt{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}}} \]

   2. pow2 97.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}}\right)}^{2}} \]

8. Applied egg-rr 97.8%

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

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. Final simplification 95.9%

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

Alternative 8: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

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)));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative 97.6%

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

   2. fma-define 97.6%

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

   3. associate-*l/ 98.3%

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

   4. *-lft-identity 98.3%

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

   5. associate--l- 98.2%

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

   6. sub-neg 98.2%

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

   7. neg-mul-1 98.2%

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

   8. distribute-lft-out 98.2%

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

   9. distribute-rgt-out 98.2%

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

   10. metadata-eval 98.2%

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

   11. distribute-lft-neg-out 98.2%

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

   12. distribute-rgt-neg-out 98.2%

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

   13. exp-prod 98.2%

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

3. Simplified 98.2%

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

   1. add-cube-cbrt 97.0%

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

   2. pow3 96.9%

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

6. Applied egg-rr 93.8%

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

7. Taylor expanded in c around 0 94.8%

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

8. Taylor expanded in cosTheta around 0 95.8%

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

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. associate-*r* 95.8%

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

   4. mul-1-neg 95.8%

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

   5. unsub-neg 95.8%

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

10. Simplified 95.8%

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

    1. sub-neg 95.8%

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

    2. pow1/2 95.8%

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

    3. inv-pow 95.8%

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

    4. pow-pow 95.8%

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

    5. metadata-eval 95.8%

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

12. Applied egg-rr 95.8%

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

    1. sub-neg 95.8%

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

14. Simplified 95.8%

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

15. Final simplification 95.8%

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

Alternative 9: 92.9% 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 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. fma-define 97.6%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

6. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\pi} \cdot cosTheta\right)\right)}}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)} \]

7. Taylor expanded in cosTheta around 0 92.6%

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

8. Final simplification 92.6%

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

Alternative 10: 5.0% 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 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. fma-define 97.6%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around inf 5.0%

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

6. Final simplification 5.0%

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

Reproduce

herbie shell --seed 2024096 
(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))))))