Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 18.4s

Alternatives: 14

Speedup: 1.5×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(1 + \frac{\sqrt{1 + cosTheta \cdot -2}}{e^{cosTheta \cdot cosTheta} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta\right)}\right) + \color{blue}{c}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{e^{cosTheta \cdot cosTheta} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta\right)} + 1\right) + c\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\sqrt{1 + cosTheta \cdot -2}}{e^{cosTheta \cdot cosTheta} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta\right)} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{e^{cosTheta \cdot cosTheta} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta\right)}\right), \color{blue}{\left(1 + c\right)}\right)\right) \]

6. Applied egg-rr 98.4%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(c + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]

7. Simplified 97.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \left(\frac{\sqrt{\frac{1 + cosTheta \cdot -2}{\mathsf{PI}\left(\right)}}}{e^{cosTheta \cdot cosTheta} \cdot \color{blue}{cosTheta}}\right)\right)\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \left(\frac{\frac{\sqrt{\frac{1 + cosTheta \cdot -2}{\mathsf{PI}\left(\right)}}}{e^{cosTheta \cdot cosTheta}}}{\color{blue}{cosTheta}}\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\left(\frac{\sqrt{\frac{1 + cosTheta \cdot -2}{\mathsf{PI}\left(\right)}}}{e^{cosTheta \cdot cosTheta}}\right), \color{blue}{cosTheta}\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\sqrt{\frac{1 + cosTheta \cdot -2}{\mathsf{PI}\left(\right)}}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\frac{\mathsf{PI}\left(\right)}{1 + cosTheta \cdot -2}}}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\sqrt{{\left(\frac{\mathsf{PI}\left(\right)}{1 + cosTheta \cdot -2}\right)}^{-1}}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\left(\frac{\mathsf{PI}\left(\right)}{1 + cosTheta \cdot -2}\right)}^{\left(\frac{-1}{2}\right)}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\left(\frac{\mathsf{PI}\left(\right)}{1 + cosTheta \cdot -2}\right)}^{\frac{-1}{2}}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   9. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{1 + cosTheta \cdot -2}\right), \frac{-1}{2}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), \left(1 + cosTheta \cdot -2\right)\right), \frac{-1}{2}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   11. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \left(1 + cosTheta \cdot -2\right)\right), \frac{-1}{2}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \left(cosTheta \cdot -2\right)\right)\right), \frac{-1}{2}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, -2\right)\right)\right), \frac{-1}{2}\right), \left(e^{cosTheta \cdot cosTheta}\right)\right), cosTheta\right)\right)\right)\right) \]

   14. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, -2\right)\right)\right), \frac{-1}{2}\right), \mathsf{exp.f32}\left(\left(cosTheta \cdot cosTheta\right)\right)\right), cosTheta\right)\right)\right)\right) \]

   15. *-lowering-*.f32 97.8%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, -2\right)\right)\right), \frac{-1}{2}\right), \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), cosTheta\right)\right)\right)\right) \]

9. Applied egg-rr 97.8%

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

Alternative 4: 98.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(c + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]

7. Simplified 97.7%

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

Alternative 5: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{\sqrt{\frac{1 - cosTheta \cdot 2}{\pi}} \cdot e^{0 - cosTheta \cdot cosTheta}}{cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (/
    (*
     (sqrt (/ (- 1.0 (* cosTheta 2.0)) PI))
     (exp (- 0.0 (* cosTheta cosTheta))))
    cosTheta))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((sqrtf(((1.0f - (cosTheta * 2.0f)) / ((float) M_PI))) * expf((0.0f - (cosTheta * cosTheta)))) / cosTheta));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(sqrt(Float32(Float32(Float32(1.0) - Float32(cosTheta * Float32(2.0))) / Float32(pi))) * exp(Float32(Float32(0.0) - Float32(cosTheta * cosTheta)))) / cosTheta)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((sqrt(((single(1.0) - (cosTheta * single(2.0))) / single(pi))) * exp((single(0.0) - (cosTheta * cosTheta)))) / cosTheta));
end

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}}\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}}{\color{blue}{cosTheta}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right), \color{blue}{cosTheta}\right)\right)\right) \]

4. Applied egg-rr 97.8%

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

5. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\color{blue}{1}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(2, cosTheta\right)\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right), cosTheta\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 97.3%

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

8. Final simplification 97.3%

   \[\leadsto \frac{1}{1 + \frac{\sqrt{\frac{1 - cosTheta \cdot 2}{\pi}} \cdot e^{0 - cosTheta \cdot cosTheta}}{cosTheta}} \]
9. Add Preprocessing

Alternative 6: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in c around 0

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

7. Simplified 97.3%

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

Alternative 7: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta \cdot \left(c \cdot c + -1\right)\\ t_1 := \left(1 - c\right) \cdot \left(1 - cosTheta\right)\\ \frac{cosTheta \cdot \left(1 - c\right)}{\frac{t\_1 \cdot t\_1}{\pi} + cosTheta \cdot \left(\left(1 - c \cdot c\right) \cdot t\_0\right)} \cdot \left(\frac{t\_1}{{\pi}^{0.5}} + t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (* cosTheta (+ (* c c) -1.0)))
        (t_1 (* (- 1.0 c) (- 1.0 cosTheta))))
   (*
    (/
     (* cosTheta (- 1.0 c))
     (+ (/ (* t_1 t_1) PI) (* cosTheta (* (- 1.0 (* c c)) t_0))))
    (+ (/ t_1 (pow PI 0.5)) t_0))))

C

float code(float cosTheta, float c) {
	float t_0 = cosTheta * ((c * c) + -1.0f);
	float t_1 = (1.0f - c) * (1.0f - cosTheta);
	return ((cosTheta * (1.0f - c)) / (((t_1 * t_1) / ((float) M_PI)) + (cosTheta * ((1.0f - (c * c)) * t_0)))) * ((t_1 / powf(((float) M_PI), 0.5f)) + t_0);
}

Julia

function code(cosTheta, c)
	t_0 = Float32(cosTheta * Float32(Float32(c * c) + Float32(-1.0)))
	t_1 = Float32(Float32(Float32(1.0) - c) * Float32(Float32(1.0) - cosTheta))
	return Float32(Float32(Float32(cosTheta * Float32(Float32(1.0) - c)) / Float32(Float32(Float32(t_1 * t_1) / Float32(pi)) + Float32(cosTheta * Float32(Float32(Float32(1.0) - Float32(c * c)) * t_0)))) * Float32(Float32(t_1 / (Float32(pi) ^ Float32(0.5))) + t_0))
end

MATLAB

function tmp = code(cosTheta, c)
	t_0 = cosTheta * ((c * c) + single(-1.0));
	t_1 = (single(1.0) - c) * (single(1.0) - cosTheta);
	tmp = ((cosTheta * (single(1.0) - c)) / (((t_1 * t_1) / single(pi)) + (cosTheta * ((single(1.0) - (c * c)) * t_0)))) * ((t_1 / (single(pi) ^ single(0.5))) + t_0);
end

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \color{blue}{\left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + -1 \cdot \left(cosTheta \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{cosTheta}\right)}\right)\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{cosTheta}\right)\right)\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(1 + -1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + -1 \cdot cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + \left(\mathsf{neg}\left(cosTheta\right)\right)\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 - cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right), cosTheta\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

   11. PI-lowering-PI.f32 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

5. Simplified 95.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \frac{1 \cdot 1 - c \cdot c}{\color{blue}{1 - c}}\right)\right) \]

   3. frac-add N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)}{\color{blue}{cosTheta \cdot \left(1 - c\right)}}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)\right), \color{blue}{\left(cosTheta \cdot \left(1 - c\right)\right)}\right)\right) \]

7. Applied egg-rr 95.1%

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

   1. clear-num N/A

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

   2. flip-+ N/A

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

   3. associate-/r/ N/A

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

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{cosTheta \cdot \left(1 - c\right)}{\left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right)\right) \cdot \left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right)\right) - \left(cosTheta \cdot \left(1 - c \cdot c\right)\right) \cdot \left(cosTheta \cdot \left(1 - c \cdot c\right)\right)}\right), \color{blue}{\left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right) - cosTheta \cdot \left(1 - c \cdot c\right)\right)}\right) \]

9. Applied egg-rr 96.2%

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

10. Final simplification 96.2%

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

Alternative 8: 95.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \color{blue}{\left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + -1 \cdot \left(cosTheta \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{cosTheta}\right)}\right)\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{cosTheta}\right)\right)\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(1 + -1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + -1 \cdot cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + \left(\mathsf{neg}\left(cosTheta\right)\right)\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 - cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right), cosTheta\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

   11. PI-lowering-PI.f32 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

5. Simplified 95.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \frac{1 \cdot 1 - c \cdot c}{\color{blue}{1 - c}}\right)\right) \]

   3. frac-add N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)}{\color{blue}{cosTheta \cdot \left(1 - c\right)}}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)\right), \color{blue}{\left(cosTheta \cdot \left(1 - c\right)\right)}\right)\right) \]

7. Applied egg-rr 95.1%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{1}{\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right) + cosTheta \cdot \left(1 - c \cdot c\right)} \cdot \left(1 - c\right)\right), \color{blue}{cosTheta}\right) \]

9. Applied egg-rr 95.9%

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

10. Final simplification 95.9%

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

Alternative 9: 95.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \color{blue}{\left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + -1 \cdot \left(cosTheta \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{cosTheta}\right)}\right)\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{cosTheta}\right)\right)\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(1 + -1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + -1 \cdot cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + \left(\mathsf{neg}\left(cosTheta\right)\right)\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 - cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right), cosTheta\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

   11. PI-lowering-PI.f32 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

5. Simplified 95.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \frac{1 \cdot 1 - c \cdot c}{\color{blue}{1 - c}}\right)\right) \]

   3. frac-add N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)}{\color{blue}{cosTheta \cdot \left(1 - c\right)}}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)\right), \color{blue}{\left(cosTheta \cdot \left(1 - c\right)\right)}\right)\right) \]

7. Applied egg-rr 95.1%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{\left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right)\right) \cdot \left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right)\right) - \left(cosTheta \cdot \left(1 - c \cdot c\right)\right) \cdot \left(cosTheta \cdot \left(1 - c \cdot c\right)\right)}{\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right) - cosTheta \cdot \left(1 - c \cdot c\right)}\right), \mathsf{*.f32}\left(\color{blue}{cosTheta}, \mathsf{\_.f32}\left(1, c\right)\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right)\right) \cdot \left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right)\right) - \left(cosTheta \cdot \left(1 - c \cdot c\right)\right) \cdot \left(cosTheta \cdot \left(1 - c \cdot c\right)\right)\right), \left(\frac{1 - cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \left(1 - c\right) - cosTheta \cdot \left(1 - c \cdot c\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{cosTheta}, \mathsf{\_.f32}\left(1, c\right)\right)\right)\right) \]

9. Applied egg-rr 95.4%

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

10. Taylor expanded in c around 0

    \[\leadsto \color{blue}{\frac{cosTheta \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(1 - cosTheta\right) - cosTheta\right)}{\frac{{\left(1 - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

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

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \color{blue}{\left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(1 - cosTheta\right) - cosTheta}{\frac{{\left(1 - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}}\right)}\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(1 - cosTheta\right) - cosTheta\right), \color{blue}{\left(\frac{{\left(1 - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}\right)}\right)\right) \]

    4. --lowering--.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(1 - cosTheta\right)\right), cosTheta\right), \left(\color{blue}{\frac{{\left(1 - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)}} - {cosTheta}^{2}\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \left(1 - cosTheta\right)\right), cosTheta\right), \left(\frac{\color{blue}{{\left(1 - cosTheta\right)}^{2}}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}\right)\right)\right) \]

    6. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(1 - cosTheta\right)\right), cosTheta\right), \left(\frac{{\color{blue}{\left(1 - cosTheta\right)}}^{2}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}\right)\right)\right) \]

    7. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(1 - cosTheta\right)\right), cosTheta\right), \left(\frac{{\left(\color{blue}{1} - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}\right)\right)\right) \]

    8. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right), \left(1 - cosTheta\right)\right), cosTheta\right), \left(\frac{{\left(1 - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}\right)\right)\right) \]

    9. --lowering--.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{\_.f32}\left(1, cosTheta\right)\right), cosTheta\right), \left(\frac{{\left(1 - cosTheta\right)}^{\color{blue}{2}}}{\mathsf{PI}\left(\right)} - {cosTheta}^{2}\right)\right)\right) \]

    10. --lowering--.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{\_.f32}\left(1, cosTheta\right)\right), cosTheta\right), \mathsf{\_.f32}\left(\left(\frac{{\left(1 - cosTheta\right)}^{2}}{\mathsf{PI}\left(\right)}\right), \color{blue}{\left({cosTheta}^{2}\right)}\right)\right)\right) \]

12. Simplified 95.8%

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

13. Final simplification 95.8%

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

Alternative 10: 95.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \color{blue}{\left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + -1 \cdot \left(cosTheta \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{cosTheta}\right)}\right)\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{cosTheta}\right)\right)\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(1 + -1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + -1 \cdot cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + \left(\mathsf{neg}\left(cosTheta\right)\right)\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 - cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right), cosTheta\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

   11. PI-lowering-PI.f32 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

5. Simplified 95.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + 1\right) + \color{blue}{c}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + 1\right), \color{blue}{c}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right), 1\right), c\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(\left(1 - cosTheta\right) \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right), 1\right), c\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(1 - cosTheta\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right), 1\right), c\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right), 1\right), c\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right), 1\right), c\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\left({\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right), cosTheta\right)\right), 1\right), c\right)\right) \]

   10. inv-pow N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\left({\left({\mathsf{PI}\left(\right)}^{-1}\right)}^{\frac{1}{2}}\right), cosTheta\right)\right), 1\right), c\right)\right) \]

   11. pow-pow N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\left({\mathsf{PI}\left(\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), cosTheta\right)\right), 1\right), c\right)\right) \]

   12. pow-lowering-pow.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{PI}\left(\right), \left(-1 \cdot \frac{1}{2}\right)\right), cosTheta\right)\right), 1\right), c\right)\right) \]

   13. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \left(-1 \cdot \frac{1}{2}\right)\right), cosTheta\right)\right), 1\right), c\right)\right) \]

   14. metadata-eval 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{-1}{2}\right), cosTheta\right)\right), 1\right), c\right)\right) \]

7. Applied egg-rr 95.1%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(\left(1 - cosTheta\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}{cosTheta} + 1\right) + c\right)}\right) \]

   2. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(1 - cosTheta\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}{cosTheta} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left(1 - cosTheta\right) \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}{cosTheta}\right), \color{blue}{\left(1 + c\right)}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left(1 - cosTheta\right) \cdot \frac{1}{\frac{cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}}\right), \left(1 + c\right)\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1 - cosTheta}{\frac{cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}}\right), \left(\color{blue}{1} + c\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(1 - cosTheta\right), \left(\frac{cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}\right)\right), \left(\color{blue}{1} + c\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\frac{cosTheta}{{\mathsf{PI}\left(\right)}^{\frac{-1}{2}}}\right)\right), \left(1 + c\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(cosTheta, \left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right)\right)\right), \left(1 + c\right)\right)\right) \]

   9. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(cosTheta, \mathsf{pow.f32}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right)\right)\right), \left(1 + c\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(cosTheta, \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{-1}{2}\right)\right)\right), \left(1 + c\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(cosTheta, \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{-1}{2}\right)\right)\right), \left(c + \color{blue}{1}\right)\right)\right) \]

   12. +-lowering-+.f32 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{/.f32}\left(cosTheta, \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{-1}{2}\right)\right)\right), \mathsf{+.f32}\left(c, \color{blue}{1}\right)\right)\right) \]

9. Applied egg-rr 95.1%

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

10. Taylor expanded in cosTheta around 0

    \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \color{blue}{\left(cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right), \mathsf{+.f32}\left(c, 1\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{*.f32}\left(cosTheta, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right), \mathsf{+.f32}\left(c, 1\right)\right)\right) \]

    2. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{*.f32}\left(cosTheta, \mathsf{sqrt.f32}\left(\mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(c, 1\right)\right)\right) \]

    3. PI-lowering-PI.f32 95.8%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{*.f32}\left(cosTheta, \mathsf{sqrt.f32}\left(\mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(c, 1\right)\right)\right) \]

12. Simplified 95.8%

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

13. Final simplification 95.8%

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

Alternative 11: 95.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \color{blue}{\left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + -1 \cdot \left(cosTheta \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{cosTheta}\right)}\right)\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \left(\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \left(-1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{cosTheta}\right)\right)\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(-1 \cdot cosTheta + 1\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\left(\left(1 + -1 \cdot cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), cosTheta\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + -1 \cdot cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 + \left(\mathsf{neg}\left(cosTheta\right)\right)\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(1 - cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right), cosTheta\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right), cosTheta\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

   11. PI-lowering-PI.f32 95.1%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{+.f32}\left(1, c\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, cosTheta\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right)\right), cosTheta\right)\right)\right) \]

5. Simplified 95.1%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \color{blue}{\left(1 + c\right)}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta} + \frac{1 \cdot 1 - c \cdot c}{\color{blue}{1 - c}}\right)\right) \]

   3. frac-add N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)}{\color{blue}{cosTheta \cdot \left(1 - c\right)}}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\left(\left(1 - cosTheta\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(1 - c\right) + cosTheta \cdot \left(1 \cdot 1 - c \cdot c\right)\right), \color{blue}{\left(cosTheta \cdot \left(1 - c\right)\right)}\right)\right) \]

7. Applied egg-rr 95.1%

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

8. Taylor expanded in c around 0

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \color{blue}{\left(cosTheta + \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(1 - cosTheta\right)\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \mathsf{+.f32}\left(cosTheta, \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(1 - cosTheta\right)\right)}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \mathsf{+.f32}\left(cosTheta, \mathsf{*.f32}\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right), \color{blue}{\left(1 - cosTheta\right)}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \mathsf{+.f32}\left(cosTheta, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right), \left(\color{blue}{1} - cosTheta\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \mathsf{+.f32}\left(cosTheta, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI}\left(\right)\right)\right), \left(1 - cosTheta\right)\right)\right)\right) \]

   6. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \mathsf{+.f32}\left(cosTheta, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right), \left(1 - cosTheta\right)\right)\right)\right) \]

   7. --lowering--.f32 95.3%

      \[\leadsto \mathsf{/.f32}\left(cosTheta, \mathsf{+.f32}\left(cosTheta, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(1, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{\_.f32}\left(1, \color{blue}{cosTheta}\right)\right)\right)\right) \]

10. Simplified 95.3%

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

11. Final simplification 95.3%

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

Alternative 12: 92.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta, \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{sqrt.f32}\left(\mathsf{PI}\left(\right)\right)\right) \]

   3. PI-lowering-PI.f32 93.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta, \mathsf{sqrt.f32}\left(\mathsf{PI.f32}\left(\right)\right)\right) \]

7. Simplified 93.1%

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

Alternative 13: 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 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(c + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]

7. Simplified 97.7%

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

8. Taylor expanded in c around inf

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{c}\right)\right) \]
9. Step-by-step derivation

10. Simplified 10.7%

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

11. Taylor expanded in c around 0

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

    1. mul-1-neg N/A

       \[\leadsto 1 + \left(\mathsf{neg}\left(c\right)\right) \]

    2. sub-neg N/A

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

    3. --lowering--.f32 10.7%

       \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{c}\right) \]

13. Simplified 10.7%

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

Alternative 14: 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 97.6%

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(c + 1\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(c + \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \color{blue}{\left(1 + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(e^{\mathsf{neg}\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{cosTheta} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(c, \mathsf{+.f32}\left(1, \left(\frac{1}{e^{cosTheta \cdot cosTheta}} \cdot \left(\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

3. Simplified 98.4%

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

5. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(1 + \left(c + \frac{1}{cosTheta \cdot e^{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}\right)\right)}\right) \]

7. Simplified 97.7%

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

8. Taylor expanded in c around inf

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \color{blue}{c}\right)\right) \]
9. Step-by-step derivation

10. Simplified 10.7%

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

11. Taylor expanded in c around 0

    \[\leadsto \color{blue}{1} \]
12. Step-by-step derivation

13. Simplified 10.7%

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

Reproduce

herbie shell --seed 2024191 
(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))))))