Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.3%

Time: 14.4s

Alternatives: 17

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: 98.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (/
     (/ (sqrt (- 1.0 (+ cosTheta cosTheta))) cosTheta)
     (sqrt (pow (cbrt PI) 3.0)))
    (exp (* cosTheta (- cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((sqrtf((1.0f - (cosTheta + cosTheta))) / cosTheta) / sqrtf(powf(cbrtf(((float) M_PI)), 3.0f))) * 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((cbrt(Float32(pi)) ^ Float32(3.0)))) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

4. Applied egg-rr 98.4%

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

   1. add-cube-cbrt 98.6%

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

   2. pow3 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * (1.0f / (sqrtf(((float) M_PI)) / (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(1.0) / Float32(sqrt(Float32(pi)) / Float32(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(1.0) / (sqrt(single(pi)) / (sqrt((single(1.0) - (cosTheta * single(2.0)))) / cosTheta)))));
end

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

   1. rem-cube-cbrt 98.4%

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

   2. associate-/r* 98.4%

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

   3. clear-num 98.4%

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

   4. count-2 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(1.0) - Float32(cosTheta + cosTheta))) / Float32(cosTheta * sqrt(Float32(pi)))))))
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 * sqrt(single(pi))))));
end

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (*
     (/ (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 + (c + ((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(c + 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) + (c + ((exp(-(cosTheta ^ single(2.0))) / cosTheta) * sqrt(((single(1.0) - (cosTheta * single(2.0))) / single(pi))))));
end

Derivation

1. Initial program 97.8%

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

   1. associate-+l+ 97.8%

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

   2. *-commutative 97.8%

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

   3. exp-prod 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.9%

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

6. Final simplification 97.9%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in c around 0 97.6%

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

   1. associate-*l/ 97.8%

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

   2. mul-1-neg 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 97.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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(1.0) + Float32(cosTheta * Float32(Float32(-1.0) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(-0.5)) - Float32(0.5)))))) / Float32(cosTheta * sqrt(Float32(pi)))))))
end

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in cosTheta around 0 97.6%

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

6. Final simplification 97.6%

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

Alternative 7: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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(1.0) + Float32(cosTheta * Float32(Float32(-1.0) + Float32(cosTheta * Float32(-0.5))))) / Float32(cosTheta * sqrt(Float32(pi)))))))
end

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in cosTheta around 0 97.3%

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

6. Final simplification 97.3%

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

Alternative 8: 96.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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(1.0) - cosTheta) / Float32(cosTheta * sqrt(Float32(pi)))))))
end

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.4%

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

   4. *-commutative 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in cosTheta around 0 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 9: 95.7% 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 97.8%

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

   1. associate-+l+ 97.8%

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

   2. *-commutative 97.8%

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

   3. exp-prod 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in cosTheta around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

   3. associate-*r* 96.4%

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

   4. mul-1-neg 96.4%

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

   5. unsub-neg 96.4%

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

7. Simplified 96.4%

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

8. Final simplification 96.4%

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

Alternative 10: 94.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{cosTheta}{t\_0 + cosTheta \cdot \left(1 - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/ cosTheta (+ t_0 (* cosTheta (- 1.0 t_0))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return cosTheta / (t_0 + (cosTheta * (1.0f - t_0)));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(cosTheta / Float32(t_0 + Float32(cosTheta * Float32(Float32(1.0) - t_0))))
end

MATLAB

function tmp = code(cosTheta, c)
	t_0 = sqrt((single(1.0) / single(pi)));
	tmp = cosTheta / (t_0 + (cosTheta * (single(1.0) - t_0)));
end

Derivation

1. Initial program 97.8%

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

   1. associate-+l+ 97.8%

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

   2. *-commutative 97.8%

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

   3. exp-prod 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

7. Simplified 95.7%

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

8. Taylor expanded in c around 0 95.8%

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

Alternative 11: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

   1. clear-num 95.8%

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

   2. inv-pow 95.8%

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

   3. pow1/2 95.8%

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

   4. inv-pow 95.8%

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

   5. pow-pow 95.8%

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

   6. metadata-eval 95.8%

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

   7. pow1/2 95.8%

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

   8. inv-pow 95.8%

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

   9. pow-pow 95.8%

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

   10. metadata-eval 95.8%

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

9. Applied egg-rr 95.8%

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

    1. unpow-1 95.8%

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

    2. cancel-sign-sub-inv 95.8%

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

    3. distribute-rgt1-in 95.8%

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

11. Simplified 95.8%

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

12. Final simplification 95.8%

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

Alternative 12: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

8. Taylor expanded in cosTheta around inf 95.6%

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

   1. distribute-rgt-out 95.6%

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

10. Simplified 95.6%

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

11. Final simplification 95.6%

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

Alternative 13: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

   1. +-commutative 95.7%

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

   2. *-un-lft-identity 95.7%

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

   3. fma-define 95.7%

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

   4. pow1/2 95.7%

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

   5. inv-pow 95.7%

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

   6. pow-pow 95.7%

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

   7. metadata-eval 95.7%

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

   8. pow1/2 95.7%

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

   9. inv-pow 95.7%

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

   10. pow-pow 95.7%

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

   11. metadata-eval 95.7%

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

9. Applied egg-rr 95.7%

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

    1. fma-undefine 95.7%

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

    2. *-lft-identity 95.7%

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

    3. cancel-sign-sub-inv 95.7%

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

    4. distribute-rgt1-in 95.7%

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

11. Simplified 95.7%

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

12. Final simplification 95.7%

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

Alternative 14: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

8. Taylor expanded in c around 0 95.5%

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

   1. associate--l+ 95.5%

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

   2. sub-neg 95.5%

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

   3. mul-1-neg 95.5%

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

   4. +-commutative 95.5%

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

   5. distribute-rgt-out 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 15: 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 97.8%

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

   1. associate-+l+ 97.8%

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

   2. *-commutative 97.8%

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

   3. exp-prod 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in cosTheta around 0 92.9%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]
6. Add Preprocessing

Alternative 16: 10.7% accurate, 107.3× 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 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in cosTheta around 0 91.4%

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

6. Taylor expanded in cosTheta around inf 10.8%

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

7. Taylor expanded in c around 0 10.8%

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

   1. neg-mul-1 10.8%

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

   2. unsub-neg 10.8%

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

9. Simplified 10.8%

   \[\leadsto \color{blue}{1 - c} \]
10. Add Preprocessing

Alternative 17: 10.7% accurate, 322.0× 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 97.8%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. frac-times 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. associate--l- 97.2%

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

   6. *-commutative 97.2%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in cosTheta around 0 91.4%

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

6. Taylor expanded in cosTheta around inf 10.8%

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

7. Taylor expanded in c around 0 10.8%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))))