Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.6%

Time: 20.4s

Alternatives: 13

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate--l- 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num 98.3%

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

   2. inv-pow 98.3%

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

   3. count-2 98.3%

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

6. Applied egg-rr 98.3%

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

   1. unpow-1 98.3%

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

   2. associate-/r/ 98.2%

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

   3. sub-neg 98.2%

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

   4. *-commutative 98.2%

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

   5. distribute-rgt-neg-in 98.2%

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

   6. metadata-eval 98.2%

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

8. Simplified 98.2%

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

   1. add0 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\left(\frac{\sqrt{\pi}}{\sqrt{1 + cosTheta \cdot -2}} + 0\right)} \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. sqrt-undiv 98.3%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\left(\color{blue}{\sqrt{\frac{\pi}{1 + cosTheta \cdot -2}}} + 0\right) \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. +-commutative 98.3%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\left(\sqrt{\frac{\pi}{\color{blue}{cosTheta \cdot -2 + 1}}} + 0\right) \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. fma-define 98.3%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\left(\sqrt{\frac{\pi}{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}} + 0\right) \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

10. Applied egg-rr 98.3%

    \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\left(\sqrt{\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}} + 0\right)} \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
11. Step-by-step derivation

    1. add0 98.3%

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

12. Simplified 98.3%

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

13. Final simplification 98.3%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate--l- 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in c around 0 97.8%

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

   1. +-commutative 97.8%

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

   2. *-commutative 97.8%

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

   3. sub-neg 97.8%

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

   4. *-commutative 97.8%

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

   5. distribute-rgt-neg-in 97.8%

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

   6. metadata-eval 97.8%

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

   7. neg-mul-1 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate--l- 98.3%

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

4. Applied egg-rr 98.3%

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

   1. clear-num 98.3%

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

   2. inv-pow 98.3%

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

   3. count-2 98.3%

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

6. Applied egg-rr 98.3%

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

   1. unpow-1 98.3%

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

   2. associate-/r/ 98.2%

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

   3. sub-neg 98.2%

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

   4. *-commutative 98.2%

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

   5. distribute-rgt-neg-in 98.2%

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

   6. metadata-eval 98.2%

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

8. Simplified 98.2%

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

   1. add0 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\left(\frac{\sqrt{\pi}}{\sqrt{1 + cosTheta \cdot -2}} + 0\right)} \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. sqrt-undiv 98.3%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\left(\color{blue}{\sqrt{\frac{\pi}{1 + cosTheta \cdot -2}}} + 0\right) \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. +-commutative 98.3%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\left(\sqrt{\frac{\pi}{\color{blue}{cosTheta \cdot -2 + 1}}} + 0\right) \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. fma-define 98.3%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\left(\sqrt{\frac{\pi}{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}} + 0\right) \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

10. Applied egg-rr 98.3%

    \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\color{blue}{\left(\sqrt{\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}} + 0\right)} \cdot cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
11. Step-by-step derivation

    1. add0 98.3%

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

12. Simplified 98.3%

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

    1. add0 98.3%

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{1}{\sqrt{\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot cosTheta} + 0\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    2. associate-/r* 97.9%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\frac{\frac{1}{\sqrt{\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}}{cosTheta}} + 0\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    3. pow1/2 97.9%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{\frac{1}{\color{blue}{{\left(\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}^{0.5}}}}{cosTheta} + 0\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    4. pow-flip 97.9%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{\color{blue}{{\left(\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}^{\left(-0.5\right)}}}{cosTheta} + 0\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    5. metadata-eval 97.9%

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{{\left(\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}^{\color{blue}{-0.5}}}{cosTheta} + 0\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

14. Applied egg-rr 97.9%

    \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{{\left(\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}^{-0.5}}{cosTheta} + 0\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
15. Step-by-step derivation

    1. add0 97.9%

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

16. Simplified 97.9%

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

17. Final simplification 97.9%

    \[\leadsto \frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{{\left(\frac{\pi}{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)}^{-0.5}}{cosTheta}} \]
18. Add Preprocessing

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-*l* 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. fma-undefine 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.1%

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

6. Final simplification 97.1%

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

Alternative 5: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate--l- 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in cosTheta around 0 95.3%

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

   1. mul-1-neg 95.3%

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

   2. unsub-neg 95.3%

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

7. Simplified 95.3%

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

8. Final simplification 95.3%

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

Alternative 6: 95.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-*l* 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. fma-undefine 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in cosTheta around 0 94.5%

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

   1. +-commutative 94.5%

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

   2. distribute-rgt-out 94.5%

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

   3. fma-define 94.6%

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

7. Simplified 94.6%

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

8. Final simplification 94.6%

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

Alternative 7: 95.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-*l* 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. fma-undefine 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.1%

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

   1. +-commutative 97.1%

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

   2. *-commutative 97.1%

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

   3. neg-mul-1 97.1%

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

   4. fma-define 97.1%

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

   5. +-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. fma-undefine 97.1%

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

   8. neg-mul-1 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.4%

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

   1. add0 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} + 0\right)} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   2. pow1/2 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   3. inv-pow 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   4. pow-pow 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   5. metadata-eval 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left({\pi}^{\color{blue}{-0.5}} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

10. Applied egg-rr 94.4%

    \[\leadsto \frac{1}{1 + \left(-1 \cdot \color{blue}{\left({\pi}^{-0.5} + 0\right)} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. add0 94.4%

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

12. Simplified 94.4%

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

    1. inv-pow 94.4%

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

    2. sqrt-div 94.4%

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

    3. metadata-eval 94.4%

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

    4. inv-pow 94.4%

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

    5. unpow-prod-down 94.9%

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

    6. inv-pow 94.9%

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

    7. *-commutative 94.9%

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

14. Applied egg-rr 94.9%

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

15. Final simplification 94.9%

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

Alternative 8: 95.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-*l* 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. fma-undefine 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.1%

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

   1. +-commutative 97.1%

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

   2. *-commutative 97.1%

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

   3. neg-mul-1 97.1%

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

   4. fma-define 97.1%

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

   5. +-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. fma-undefine 97.1%

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

   8. neg-mul-1 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.4%

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

   1. add0 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} + 0\right)} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   2. pow1/2 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   3. inv-pow 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   4. pow-pow 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   5. metadata-eval 94.4%

      \[\leadsto \frac{1}{1 + \left(-1 \cdot \left({\pi}^{\color{blue}{-0.5}} + 0\right) + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

10. Applied egg-rr 94.4%

    \[\leadsto \frac{1}{1 + \left(-1 \cdot \color{blue}{\left({\pi}^{-0.5} + 0\right)} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. add0 94.4%

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

12. Simplified 94.4%

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

    1. inv-pow 94.4%

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

    2. sqrt-div 94.4%

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

    3. metadata-eval 94.4%

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

    4. inv-pow 94.4%

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

    5. unpow-prod-down 94.9%

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

    6. inv-pow 94.9%

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

    7. associate-/r* 94.9%

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

14. Applied egg-rr 94.9%

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

15. Final simplification 94.9%

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

Alternative 9: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate--l- 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in cosTheta around 0 94.5%

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

   1. associate-+r+ 94.5%

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

   2. distribute-rgt-out 94.5%

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

7. Simplified 94.5%

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

8. Final simplification 94.5%

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

Alternative 10: 94.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-*l* 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. fma-undefine 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.1%

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

   1. +-commutative 97.1%

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

   2. *-commutative 97.1%

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

   3. neg-mul-1 97.1%

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

   4. fma-define 97.1%

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

   5. +-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. fma-undefine 97.1%

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

   8. neg-mul-1 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in cosTheta around 0 94.4%

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

   1. expm1-log1p-u 87.5%

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

   2. log1p-define 87.5%

      \[\leadsto \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(-1 \cdot \sqrt{\frac{1}{\pi}} + \frac{1}{cosTheta} \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right)} \]

   3. expm1-undefine 87.4%

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

   4. add-exp-log 94.4%

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

   5. +-commutative 94.4%

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

   6. distribute-rgt-out 94.4%

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

   7. fma-define 94.4%

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

   8. pow1/2 94.4%

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

   9. inv-pow 94.4%

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

   10. pow-pow 94.4%

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

   11. metadata-eval 94.4%

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

10. Applied egg-rr 94.4%

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

    1. fma-undefine 94.4%

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

    2. associate--l+ 94.4%

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

    3. metadata-eval 94.4%

       \[\leadsto \frac{1}{1 + \left({\pi}^{-0.5} \cdot \left(-1 + \frac{1}{cosTheta}\right) + \color{blue}{0}\right)} \]

    4. add0 94.4%

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

12. Simplified 94.4%

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

13. Final simplification 94.4%

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

Alternative 11: 93.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. +-commutative 97.8%

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

   4. associate-*l* 97.8%

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

   5. *-commutative 97.8%

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

   6. associate-*l* 97.8%

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

   7. fma-undefine 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in cosTheta around 0 92.8%

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

6. Final simplification 92.8%

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

Alternative 12: 10.7% accurate, 107.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 96.9%

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

   2. pow3 97.0%

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

4. Applied egg-rr 96.4%

   \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\mathsf{fma}\left(e^{-{cosTheta}^{2}}, \frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta \cdot \sqrt{\pi}}, 1 + c\right)}}\right)}^{3}} \]

5. Taylor expanded in cosTheta around 0 90.5%

   \[\leadsto {\left(\frac{1}{\sqrt[3]{\mathsf{fma}\left(e^{-{cosTheta}^{2}}, \frac{\color{blue}{1}}{cosTheta \cdot \sqrt{\pi}}, 1 + c\right)}}\right)}^{3} \]

6. Taylor expanded in cosTheta around inf 11.2%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \frac{1}{1 + c}} \]
7. Step-by-step derivation

   1. pow-base-1 11.2%

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

   2. associate-*r/ 11.2%

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

   3. metadata-eval 11.2%

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

8. Simplified 11.2%

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

9. Taylor expanded in c around 0 11.2%

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

    1. mul-1-neg 11.2%

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

    2. unsub-neg 11.2%

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

11. Simplified 11.2%

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

12. Final simplification 11.2%

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

Alternative 13: 10.7% accurate, 322.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 96.9%

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

   2. pow3 97.0%

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

4. Applied egg-rr 96.4%

   \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\mathsf{fma}\left(e^{-{cosTheta}^{2}}, \frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{cosTheta \cdot \sqrt{\pi}}, 1 + c\right)}}\right)}^{3}} \]

5. Taylor expanded in cosTheta around 0 90.5%

   \[\leadsto {\left(\frac{1}{\sqrt[3]{\mathsf{fma}\left(e^{-{cosTheta}^{2}}, \frac{\color{blue}{1}}{cosTheta \cdot \sqrt{\pi}}, 1 + c\right)}}\right)}^{3} \]

6. Taylor expanded in cosTheta around inf 11.2%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \frac{1}{1 + c}} \]
7. Step-by-step derivation

   1. pow-base-1 11.2%

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

   2. associate-*r/ 11.2%

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

   3. metadata-eval 11.2%

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

8. Simplified 11.2%

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

9. Taylor expanded in c around 0 11.2%

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

10. Final simplification 11.2%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))))))