Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 14.1s

Alternatives: 13

Speedup: 1.3×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. distribute-lft-neg-out 98.0%

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

   3. exp-neg 98.0%

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

   4. sqr-neg 98.0%

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

   5. associate-*r/ 98.0%

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

   6. associate-/l* 98.0%

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

   7. /-rgt-identity 98.0%

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

   8. *-commutative 98.0%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 0.8× speedup

Math

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

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/l* 98.7%

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

   6. /-rgt-identity 98.7%

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

   7. associate-/l* 98.7%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((sqrtf((1.0f - (cosTheta + cosTheta))) / 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(Float32(sqrt(Float32(Float32(1.0) - Float32(cosTheta + cosTheta))) / cosTheta) / sqrt(Float32(pi))) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--l- 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 4: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * (sqrtf(((1.0f - (cosTheta + cosTheta)) / ((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(Float32(Float32(1.0) - Float32(cosTheta + cosTheta)) / Float32(pi))) / cosTheta))))
end

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate-/r* 98.1%

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

   4. associate--l- 98.1%

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

3. Applied egg-rr 98.1%

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

   1. sqrt-undiv 98.2%

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

5. Applied egg-rr 98.2%

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

6. Final simplification 98.2%

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

Alternative 5: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/r/ 98.6%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.5%

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

4. Taylor expanded in c around 0 96.9%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}} \cdot \frac{1}{e^{{cosTheta}^{2}} \cdot cosTheta} + 1}} \]
5. Step-by-step derivation

   1. un-div-inv 96.9%

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

   2. *-commutative 96.9%

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

   3. *-commutative 96.9%

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

   4. unpow2 96.9%

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

6. Applied egg-rr 96.9%

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

7. Final simplification 96.9%

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

Alternative 6: 96.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/l* 98.7%

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

   6. /-rgt-identity 98.7%

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

   7. associate-/l* 98.7%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 96.7%

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

   1. distribute-rgt1-in 96.7%

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

   2. unpow2 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in cosTheta around 0 95.7%

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

   1. neg-mul-1 94.1%

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

   2. associate-+r+ 94.1%

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

   3. sub-neg 94.1%

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

   4. *-commutative 94.1%

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

   5. unpow2 94.1%

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

9. Simplified 95.7%

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

10. Final simplification 95.7%

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

Alternative 7: 95.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/l* 98.7%

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

   6. /-rgt-identity 98.7%

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

   7. associate-/l* 98.7%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 94.1%

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

5. Taylor expanded in cosTheta around 0 94.1%

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

   1. neg-mul-1 94.1%

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

   2. associate-+r+ 94.1%

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

   3. sub-neg 94.1%

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

   4. *-commutative 94.1%

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

   5. unpow2 94.1%

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

7. Simplified 94.1%

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

8. Final simplification 94.1%

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

Alternative 8: 95.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/l* 98.7%

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

   6. /-rgt-identity 98.7%

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

   7. associate-/l* 98.7%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 96.7%

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

   1. distribute-rgt1-in 96.7%

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

   2. unpow2 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in cosTheta around 0 94.3%

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

   1. neg-mul-1 94.3%

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

   2. sub-neg 94.3%

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

9. Simplified 94.3%

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

10. Final simplification 94.3%

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

Alternative 9: 95.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/r/ 98.6%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 94.3%

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

   1. neg-mul-1 94.3%

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

   2. sub-neg 94.3%

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

6. Simplified 94.3%

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

7. Taylor expanded in cosTheta around 0 93.8%

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

8. Final simplification 93.8%

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

Alternative 10: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate-/l* 98.7%

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

   6. /-rgt-identity 98.7%

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

   7. associate-/l* 98.7%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 94.1%

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

5. Taylor expanded in cosTheta around 0 93.9%

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

   1. neg-mul-1 94.3%

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

   2. sub-neg 94.3%

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

7. Simplified 93.9%

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

8. Final simplification 93.9%

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

Alternative 11: 92.9% accurate, 2.1× 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 98.0%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--l- 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in cosTheta around 0 91.3%

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

5. Final simplification 91.3%

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

Alternative 12: 10.8% accurate, 140.3× speedup

Math

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

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 98.0%

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

   1. associate-+l+ 98.0%

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

   2. distribute-lft-neg-out 98.0%

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

   3. exp-neg 98.0%

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

   4. sqr-neg 98.0%

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

   5. associate-*r/ 98.0%

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

   6. associate-/l* 98.0%

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

   7. /-rgt-identity 98.0%

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

   8. *-commutative 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 11.3%

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

5. Taylor expanded in c around 0 11.3%

   \[\leadsto \color{blue}{-1 \cdot c + 1} \]
6. Step-by-step derivation

   1. +-commutative 11.3%

      \[\leadsto \color{blue}{1 + -1 \cdot c} \]

   2. mul-1-neg 11.3%

      \[\leadsto 1 + \color{blue}{\left(-c\right)} \]

   3. unsub-neg 11.3%

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

7. Simplified 11.3%

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

8. Final simplification 11.3%

   \[\leadsto 1 - c \]

Alternative 13: 10.8% accurate, 421.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. distribute-lft-neg-out 98.0%

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

   3. exp-neg 98.0%

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

   4. sqr-neg 98.0%

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

   5. associate-*r/ 98.0%

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

   6. associate-/l* 98.0%

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

   7. /-rgt-identity 98.0%

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

   8. *-commutative 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 11.3%

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

5. Taylor expanded in c around 0 11.3%

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

6. Final simplification 11.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023268 
(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))))))