Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 15.3s

Alternatives: 9

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: 97.9% 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. clear-num 98.5%

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

   4. associate--l- 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 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 97.7%

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

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. associate-/r* 97.9%

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

   4. associate--l- 97.9%

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

3. Applied egg-rr 97.9%

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

   1. sqrt-undiv 97.9%

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

5. Applied egg-rr 97.9%

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

6. Final simplification 97.9%

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

Alternative 3: 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. associate-*l* 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.4%

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

   6. /-rgt-identity 98.4%

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

   7. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 96.8%

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

   1. neg-mul-1 96.8%

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

   2. associate-+r+ 96.8%

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

   3. sub-neg 96.8%

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

   4. *-commutative 96.8%

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

   5. unpow2 96.8%

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

6. Simplified 96.8%

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

7. Taylor expanded in cosTheta around 0 96.8%

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

   1. distribute-rgt1-in 96.7%

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

   2. unpow2 96.7%

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

9. Simplified 96.7%

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

10. Final simplification 96.7%

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

Alternative 4: 95.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. associate-*l* 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.4%

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

   6. /-rgt-identity 98.4%

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

   7. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.8%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

6. Final simplification 95.7%

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

Alternative 5: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. associate-*l* 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.4%

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

   6. /-rgt-identity 98.4%

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

   7. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.8%

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

5. Taylor expanded in cosTheta around 0 95.6%

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

6. Final simplification 95.6%

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

Alternative 6: 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. associate-*l* 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.4%

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

   6. /-rgt-identity 98.4%

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

   7. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.8%

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

5. Taylor expanded in cosTheta around 0 95.6%

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

   1. neg-mul-1 95.6%

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

   2. sub-neg 95.6%

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

7. Simplified 95.6%

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

8. Final simplification 95.6%

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

Alternative 7: 92.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. clear-num 98.5%

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

   4. associate--l- 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in cosTheta around 0 93.8%

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

5. Final simplification 93.8%

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

Alternative 8: 10.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. associate-*l* 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.4%

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

   6. associate-/l/ 98.4%

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

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 95.8%

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

   1. fma-def 95.8%

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

   2. associate-*r* 95.8%

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

   3. mul-1-neg 95.8%

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

   4. unpow2 95.8%

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

   5. *-commutative 95.8%

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

   6. mul-1-neg 95.8%

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

6. Simplified 95.8%

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

7. Taylor expanded in c around inf 11.2%

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

   1. mul-1-neg 11.2%

      \[\leadsto \color{blue}{-c \cdot \left({cosTheta}^{2} \cdot \pi\right)} \]

   2. *-commutative 11.2%

      \[\leadsto -\color{blue}{\left({cosTheta}^{2} \cdot \pi\right) \cdot c} \]

   3. distribute-rgt-neg-in 11.2%

      \[\leadsto \color{blue}{\left({cosTheta}^{2} \cdot \pi\right) \cdot \left(-c\right)} \]

   4. *-commutative 11.2%

      \[\leadsto \color{blue}{\left(\pi \cdot {cosTheta}^{2}\right)} \cdot \left(-c\right) \]

   5. unpow2 11.2%

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

9. Simplified 11.2%

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

10. Final simplification 11.2%

    \[\leadsto c \cdot \left(\left(cosTheta \cdot cosTheta\right) \cdot \left(-\pi\right)\right) \]

Alternative 9: 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. associate-*l* 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.4%

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

   6. associate-/l/ 98.4%

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

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 inf 10.3%

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

5. Taylor expanded in c around 0 10.3%

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

   1. neg-mul-1 10.3%

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

   2. +-commutative 10.3%

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

   3. unsub-neg 10.3%

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

7. Simplified 10.3%

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

8. Final simplification 10.3%

   \[\leadsto 1 - c \]

Reproduce

herbie shell --seed 2023274 
(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))))))