Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.6%

Time: 16.1s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 2: 98.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-/r* 97.7%

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

   3. sqrt-undiv 97.7%

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

   4. +-commutative 97.7%

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

   5. fma-define 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

8. Simplified 97.7%

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

Alternative 3: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-/r* 97.7%

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

   3. sqrt-undiv 97.7%

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

   4. +-commutative 97.7%

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

   5. fma-define 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

8. Simplified 97.7%

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

   1. fma-undefine 97.7%

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

   2. pow-exp 97.7%

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

   3. distribute-lft-neg-out 97.7%

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

   4. unpow2 97.7%

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

10. Applied egg-rr 97.7%

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

11. Final simplification 97.7%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-/r* 97.7%

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

   3. sqrt-undiv 97.7%

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

   4. +-commutative 97.7%

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

   5. fma-define 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in cosTheta around inf 97.7%

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

    1. associate-*r/ 97.7%

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

    2. metadata-eval 97.7%

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

11. Simplified 97.7%

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

    1. fma-undefine 97.7%

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

    2. associate-/r* 97.4%

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

    3. pow-exp 97.4%

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

    4. distribute-lft-neg-out 97.4%

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

    5. unpow2 97.4%

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

13. Applied egg-rr 97.4%

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

    1. associate-/l/ 97.7%

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

    2. inv-pow 97.7%

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

15. Applied egg-rr 97.7%

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

    1. unpow-1 97.7%

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

17. Simplified 97.7%

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

18. Final simplification 97.7%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. *-un-lft-identity 97.2%

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

   2. pow1/2 97.2%

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

   3. pow-flip 97.2%

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

   4. metadata-eval 97.2%

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

4. Applied egg-rr 97.2%

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

   1. *-lft-identity 97.2%

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

6. Simplified 97.2%

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

   1. associate-*r/ 97.5%

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

   2. associate--l- 97.5%

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

8. Applied egg-rr 97.5%

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

9. Final simplification 97.5%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-/r* 97.7%

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

   3. sqrt-undiv 97.7%

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

   4. +-commutative 97.7%

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

   5. fma-define 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in cosTheta around inf 97.7%

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

    1. associate-*r/ 97.7%

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

    2. metadata-eval 97.7%

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

11. Simplified 97.7%

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

    1. fma-undefine 97.7%

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

    2. associate-/r* 97.4%

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

    3. pow-exp 97.4%

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

    4. distribute-lft-neg-out 97.4%

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

    5. unpow2 97.4%

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

13. Applied egg-rr 97.4%

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

    1. sub-div 97.4%

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

15. Applied egg-rr 97.4%

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

16. Final simplification 97.4%

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

Alternative 7: 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 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

   1. *-un-lft-identity 98.3%

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

   2. associate-/r* 97.7%

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

   3. sqrt-undiv 97.7%

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

   4. +-commutative 97.7%

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

   5. fma-define 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in c around 0 97.4%

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

    1. neg-mul-1 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 8: 96.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. +-commutative 97.2%

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

   2. fma-define 97.2%

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

   3. associate-*l/ 97.8%

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

   4. *-lft-identity 97.8%

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

   5. sub-neg 97.8%

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

   6. associate--l+ 97.8%

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

   7. unsub-neg 97.8%

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

   8. neg-mul-1 97.8%

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

   9. neg-mul-1 97.8%

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

   10. distribute-rgt-out 97.8%

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

   11. metadata-eval 97.8%

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

   12. *-commutative 97.8%

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

   13. exp-prod 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in cosTheta around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in c around 0 96.0%

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

   1. associate-+r+ 96.0%

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

10. Simplified 96.0%

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

11. Final simplification 96.0%

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

Alternative 9: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

3. Simplified 98.3%

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

5. Taylor expanded in cosTheta around 0 94.9%

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

   1. mul-1-neg 94.9%

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

   2. unsub-neg 94.9%

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

   3. associate-*r* 94.9%

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

   4. *-commutative 94.9%

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

   5. mul-1-neg 94.9%

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

   6. unsub-neg 94.9%

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

7. Simplified 94.9%

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

   1. pow1 94.9%

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

   2. associate-*l* 94.9%

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

   3. pow1/2 94.9%

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

   4. inv-pow 94.9%

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

   5. pow-pow 94.9%

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

   6. metadata-eval 94.9%

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

9. Applied egg-rr 94.9%

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

    1. unpow1 94.9%

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

11. Simplified 94.9%

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

12. Final simplification 94.9%

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

Reproduce

herbie shell --seed 2024179 
(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))))))