Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 15.5s

Alternatives: 12

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: 98.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (fma
    (/ (sqrt (fma cosTheta -2.0 1.0)) (* cosTheta (sqrt PI)))
    (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)) / (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(fma(cosTheta, Float32(-2.0), Float32(1.0))) / Float32(cosTheta * sqrt(Float32(pi)))), (exp(Float32(-cosTheta)) ^ cosTheta), c)))
end

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. fma-def 97.6%

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

3. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 97.5%

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

   1. distribute-lft-neg-out 97.5%

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

   2. exp-neg 97.5%

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

   3. associate-*r/ 97.5%

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

   4. associate-/l* 97.5%

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

   5. /-rgt-identity 97.5%

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

   6. *-commutative 97.5%

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

3. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

6. Applied egg-rr 98.3%

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

   1. expm1-log1p-u 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{e^{{cosTheta}^{2}}}{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}}\right)\right)}} \]

   2. expm1-udef 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{e^{{cosTheta}^{2}}}{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}}\right)} - 1\right)}} \]

   3. associate-/r/ 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{e^{{cosTheta}^{2}}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}\right)} - 1\right)} \]

   4. pow2 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\frac{1}{e^{\color{blue}{cosTheta \cdot cosTheta}}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)} \]

   5. rec-exp 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{e^{-cosTheta \cdot cosTheta}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)} \]

   6. pow2 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(e^{-\color{blue}{{cosTheta}^{2}}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)} \]

8. Applied egg-rr 90.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-{cosTheta}^{2}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)}} \]
9. Step-by-step derivation

   1. expm1-def 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-{cosTheta}^{2}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)\right)}} \]

   2. expm1-log1p 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in c around 0 97.1%

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

12. Final simplification 97.1%

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

Alternative 4: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(\left(cosTheta \cdot -1.5 + \left({cosTheta}^{2} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) + -1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (pow PI -0.5)
    (+
     (+ (* cosTheta -1.5) (+ (* (pow cosTheta 2.0) 0.5) (/ 1.0 cosTheta)))
     -1.0)))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (powf(((float) M_PI), -0.5f) * (((cosTheta * -1.5f) + ((powf(cosTheta, 2.0f) * 0.5f) + (1.0f / cosTheta))) + -1.0f)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32((Float32(pi) ^ Float32(-0.5)) * Float32(Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32((cosTheta ^ Float32(2.0)) * Float32(0.5)) + Float32(Float32(1.0) / cosTheta))) + Float32(-1.0)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((single(pi) ^ single(-0.5)) * (((cosTheta * single(-1.5)) + (((cosTheta ^ single(2.0)) * single(0.5)) + (single(1.0) / cosTheta))) + single(-1.0))));
end

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 96.6%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\pi}}\right)\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   2. expm1-udef 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   3. pow1/2 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\pi}^{0.5}}}\right)} - 1\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   4. pow-flip 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(-0.5\right)}}\right)} - 1\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   5. metadata-eval 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

7. Applied egg-rr 96.6%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5}\right)\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   2. expm1-log1p 96.6%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

9. Simplified 96.6%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

10. Final simplification 96.6%

    \[\leadsto \frac{1}{\left(1 + c\right) + {\pi}^{-0.5} \cdot \left(\left(cosTheta \cdot -1.5 + \left({cosTheta}^{2} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) + -1\right)} \]
11. Add Preprocessing

Alternative 5: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\pi}^{-0.5} \cdot \left(\left(cosTheta \cdot -1.5 + \left({cosTheta}^{2} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) + -1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (pow PI -0.5)
    (+
     (+ (* cosTheta -1.5) (+ (* (pow cosTheta 2.0) 0.5) (/ 1.0 cosTheta)))
     -1.0)))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (powf(((float) M_PI), -0.5f) * (((cosTheta * -1.5f) + ((powf(cosTheta, 2.0f) * 0.5f) + (1.0f / cosTheta))) + -1.0f)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((Float32(pi) ^ Float32(-0.5)) * Float32(Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32((cosTheta ^ Float32(2.0)) * Float32(0.5)) + Float32(Float32(1.0) / cosTheta))) + Float32(-1.0)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(pi) ^ single(-0.5)) * (((cosTheta * single(-1.5)) + (((cosTheta ^ single(2.0)) * single(0.5)) + (single(1.0) / cosTheta))) + single(-1.0))));
end

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 96.6%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]

6. Taylor expanded in c around 0 96.2%

   \[\leadsto \color{blue}{\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 96.2%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   2. sqrt-div 96.2%

      \[\leadsto \frac{1}{1 + \left(1 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   3. metadata-eval 96.2%

      \[\leadsto \frac{1}{1 + \left(1 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   4. pow1/2 96.2%

      \[\leadsto \frac{1}{1 + \left(1 \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   5. pow-flip 96.2%

      \[\leadsto \frac{1}{1 + \left(1 \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

   6. metadata-eval 96.2%

      \[\leadsto \frac{1}{1 + \left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

8. Applied egg-rr 96.2%

   \[\leadsto \frac{1}{1 + \color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]
9. Step-by-step derivation

   1. *-lft-identity 96.2%

      \[\leadsto \frac{1}{1 + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

10. Simplified 96.2%

    \[\leadsto \frac{1}{1 + \color{blue}{{\pi}^{-0.5}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)} \]

11. Final simplification 96.2%

    \[\leadsto \frac{1}{1 + {\pi}^{-0.5} \cdot \left(\left(cosTheta \cdot -1.5 + \left({cosTheta}^{2} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) + -1\right)} \]
12. Add Preprocessing

Alternative 6: 96.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(cosTheta \cdot -1.5 + \frac{1}{cosTheta}\right) + -1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (* (/ 1.0 (sqrt PI)) (+ (+ (* cosTheta -1.5) (/ 1.0 cosTheta)) -1.0)))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((1.0f / sqrtf(((float) M_PI))) * (((cosTheta * -1.5f) + (1.0f / cosTheta)) + -1.0f)));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32(1.0) / cosTheta)) + Float32(-1.0)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((single(1.0) / sqrt(single(pi))) * (((cosTheta * single(-1.5)) + (single(1.0) / cosTheta)) + single(-1.0))));
end

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 95.7%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \frac{1}{cosTheta}\right) - 1\right)}} \]

6. Final simplification 95.7%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\left(cosTheta \cdot -1.5 + \frac{1}{cosTheta}\right) + -1\right)} \]
7. Add Preprocessing

Alternative 7: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(\left(cosTheta \cdot -1.5 + \frac{1}{cosTheta}\right) + -1\right) \cdot \sqrt{\frac{1}{\pi}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (* (+ (+ (* cosTheta -1.5) (/ 1.0 cosTheta)) -1.0) (sqrt (/ 1.0 PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((((cosTheta * -1.5f) + (1.0f / cosTheta)) + -1.0f) * sqrtf((1.0f / ((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32(1.0) / cosTheta)) + Float32(-1.0)) * sqrt(Float32(Float32(1.0) / Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((((cosTheta * single(-1.5)) + (single(1.0) / cosTheta)) + single(-1.0)) * sqrt((single(1.0) / single(pi)))));
end

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 96.6%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]

6. Taylor expanded in c around 0 96.2%

   \[\leadsto \color{blue}{\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]

7. Taylor expanded in cosTheta around 0 95.4%

   \[\leadsto \frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(\left(-1.5 \cdot cosTheta + \color{blue}{\frac{1}{cosTheta}}\right) - 1\right)} \]

8. Final simplification 95.4%

   \[\leadsto \frac{1}{1 + \left(\left(cosTheta \cdot -1.5 + \frac{1}{cosTheta}\right) + -1\right) \cdot \sqrt{\frac{1}{\pi}}} \]
9. Add Preprocessing

Alternative 8: 95.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 94.0%

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

6. Final simplification 94.0%

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

Alternative 9: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in cosTheta around 0 94.8%

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

   1. distribute-rgt-out 94.8%

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

   2. unpow2 94.8%

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

   3. distribute-rgt1-in 94.8%

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

9. Simplified 94.8%

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

10. Final simplification 94.8%

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

Alternative 10: 92.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. *-lft-identity 97.6%

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

   4. associate--l- 97.5%

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

   5. *-commutative 97.5%

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

   6. exp-prod 97.5%

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

3. Simplified 97.5%

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

6. Applied egg-rr 98.3%

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

   1. expm1-log1p-u 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{e^{{cosTheta}^{2}}}{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}}\right)\right)}} \]

   2. expm1-udef 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{e^{{cosTheta}^{2}}}{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}}\right)} - 1\right)}} \]

   3. associate-/r/ 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{e^{{cosTheta}^{2}}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}\right)} - 1\right)} \]

   4. pow2 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\frac{1}{e^{\color{blue}{cosTheta \cdot cosTheta}}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)} \]

   5. rec-exp 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{e^{-cosTheta \cdot cosTheta}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)} \]

   6. pow2 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(e^{\mathsf{log1p}\left(e^{-\color{blue}{{cosTheta}^{2}}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)} \]

8. Applied egg-rr 90.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{-{cosTheta}^{2}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)} - 1\right)}} \]
9. Step-by-step derivation

   1. expm1-def 90.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-{cosTheta}^{2}} \cdot \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}\right)\right)}} \]

   2. expm1-log1p 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in cosTheta around 0 92.2%

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

12. Final simplification 92.2%

    \[\leadsto cosTheta \cdot \sqrt{\pi} \]
13. Add Preprocessing

Alternative 11: 10.8% accurate, 140.3× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 (- 1.0 c))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. fma-def 97.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta around inf 10.7%

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

6. Taylor expanded in c around 0 10.7%

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

   1. mul-1-neg 10.7%

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

   2. unsub-neg 10.7%

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

8. Simplified 10.7%

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

9. Final simplification 10.7%

   \[\leadsto 1 - c \]
10. Add Preprocessing

Alternative 12: 10.8% accurate, 421.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. distribute-lft-neg-out 97.5%

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

   2. exp-neg 97.5%

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

   3. associate-*r/ 97.5%

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

   4. associate-/l* 97.5%

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

   5. /-rgt-identity 97.5%

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

   6. *-commutative 97.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in c around 0 97.1%

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

6. Taylor expanded in cosTheta around inf 10.7%

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

7. Final simplification 10.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))