Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.4%

Time: 23.5s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. *-lft-identity 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-lft-identity 97.7%

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

   5. associate--l- 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 97.7%

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

   1. associate--r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*l* 97.7%

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

   4. frac-times 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. associate--r+ 98.4%

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

   7. associate-/l/ 98.3%

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

   8. *-commutative 98.3%

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

   9. pow-exp 98.3%

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

   10. pow-neg 98.3%

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

   11. div-inv 98.3%

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

   12. associate-/l/ 98.3%

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

   13. pow-exp 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 2: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. *-lft-identity 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-lft-identity 97.7%

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

   5. associate--l- 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 97.7%

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

   1. associate--r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*l* 97.7%

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

   4. frac-times 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. associate--r+ 98.4%

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

   7. associate-/l/ 98.3%

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

   8. *-commutative 98.3%

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

   9. pow-exp 98.3%

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

   10. pow-neg 98.3%

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

   11. div-inv 98.3%

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

   12. clear-num 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in c around 0 97.5%

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

   1. associate-*r/ 97.5%

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

   2. *-rgt-identity 97.5%

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

   3. sub-neg 97.5%

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

   4. *-commutative 97.5%

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

   5. distribute-rgt-neg-in 97.5%

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

   6. metadata-eval 97.5%

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

   7. *-commutative 97.5%

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

   8. unpow2 97.5%

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 3: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. *-lft-identity 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-lft-identity 97.7%

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

   5. associate--l- 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 97.7%

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

   1. associate--r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*l* 97.7%

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

   4. frac-times 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. associate--r+ 98.4%

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

   7. associate-/l/ 98.3%

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

   8. *-commutative 98.3%

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

   9. pow-exp 98.3%

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

   10. pow-neg 98.3%

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

   11. div-inv 98.3%

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

   12. clear-num 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in cosTheta around 0 95.5%

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

   1. distribute-rgt-out 95.5%

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

   2. unpow2 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

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

Alternative 4: 95.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   2. *-lft-identity 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-lft-identity 97.7%

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

   5. associate--l- 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 97.7%

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

   1. associate--r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*l* 97.7%

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

   4. frac-times 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. associate--r+ 98.4%

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

   7. associate-/l/ 98.3%

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

   8. *-commutative 98.3%

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

   9. pow-exp 98.3%

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

   10. pow-neg 98.3%

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

   11. div-inv 98.3%

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

   12. clear-num 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in c around 0 97.5%

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

   1. cancel-sign-sub-inv 97.5%

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

   2. metadata-eval 97.5%

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

   3. associate-/r* 97.5%

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

   4. rec-exp 97.5%

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

   5. unpow2 97.5%

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

8. Simplified 97.5%

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

9. Taylor expanded in cosTheta around 0 95.0%

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

    1. distribute-rgt-out 94.9%

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

11. Simplified 94.9%

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

12. Final simplification 94.9%

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

Alternative 5: 93.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. *-lft-identity 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-lft-identity 97.7%

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

   5. associate--l- 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 97.7%

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

   1. associate--r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*l* 97.7%

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

   4. frac-times 98.4%

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

   5. *-un-lft-identity 98.4%

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

   6. associate--r+ 98.4%

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

   7. associate-/l/ 98.3%

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

   8. *-commutative 98.3%

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

   9. pow-exp 98.3%

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

   10. pow-neg 98.3%

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

   11. div-inv 98.3%

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

   12. associate-/l/ 98.3%

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

   13. pow-exp 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in cosTheta around 0 93.9%

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

7. Final simplification 93.9%

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

Alternative 6: 10.7% accurate, 421.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. +-commutative 97.7%

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

   2. associate-+l+ 97.7%

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

   3. distribute-lft-neg-out 97.7%

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

   4. exp-neg 97.8%

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

   5. associate-*r/ 97.8%

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

   6. associate-/l* 97.8%

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

   7. /-rgt-identity 97.8%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around inf 10.6%

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

5. Taylor expanded in c around 0 10.6%

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

6. Final simplification 10.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(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))))))