Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 14.3s

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(sqrt(Float32(pi)) * Float32(cosTheta * exp(Float32(cosTheta * cosTheta))))))))
end

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

Alternative 2: 98.1% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 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(Float32(1.0) + 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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

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

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

9. Simplified 98.0%

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

Alternative 3: 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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

5. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 4: 97.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 98.0%

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

7. Simplified 98.0%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. pow1/2 N/A

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

   4. *-commutative N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. pow1/2 N/A

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

   9. *-commutative N/A

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

   10. associate-/l/ N/A

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

   11. clear-num N/A

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

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

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

9. Applied egg-rr 98.0%

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

Alternative 5: 97.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in c around 0

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

7. Simplified 98.3%

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

8. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\color{blue}{\left(\frac{1 + cosTheta \cdot \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right) - 1\right)}{cosTheta}\right)}, \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(1 + cosTheta \cdot \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right) - 1\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(cosTheta \cdot \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right) - 1\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right) - 1\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right) + -1\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(-1 + cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \left(cosTheta \cdot \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \left(\frac{1}{2} \cdot cosTheta - \frac{3}{2}\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \left(\frac{1}{2} \cdot cosTheta + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \left(\frac{1}{2} \cdot cosTheta + \frac{-3}{2}\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \left(\frac{-3}{2} + \frac{1}{2} \cdot cosTheta\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(\frac{-3}{2}, \left(\frac{1}{2} \cdot cosTheta\right)\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(\frac{-3}{2}, \left(cosTheta \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   14. *-lowering-*.f32 97.9%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(\frac{-3}{2}, \mathsf{*.f32}\left(cosTheta, \frac{1}{2}\right)\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

10. Simplified 97.9%

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

Alternative 6: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*l/ N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

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

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

10. Simplified 97.5%

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

Alternative 7: 96.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*l/ N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

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

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

10. Simplified 97.4%

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

Alternative 8: 96.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in c around 0

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

7. Simplified 98.3%

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

8. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\color{blue}{\left(\frac{1 + cosTheta \cdot \left(\frac{-3}{2} \cdot cosTheta - 1\right)}{cosTheta}\right)}, \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(1 + cosTheta \cdot \left(\frac{-3}{2} \cdot cosTheta - 1\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(cosTheta \cdot \left(\frac{-3}{2} \cdot cosTheta - 1\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(\frac{-3}{2} \cdot cosTheta - 1\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(\frac{-3}{2} \cdot cosTheta + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(\frac{-3}{2} \cdot cosTheta + -1\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \left(-1 + \frac{-3}{2} \cdot cosTheta\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \left(\frac{-3}{2} \cdot cosTheta\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \left(cosTheta \cdot \frac{-3}{2}\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

   9. *-lowering-*.f32 97.4%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(cosTheta, \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \frac{-3}{2}\right)\right)\right)\right), cosTheta\right), \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

10. Simplified 97.4%

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

Alternative 9: 95.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{{\pi}^{0.5}}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 95.5%

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

5. Simplified 95.5%

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

6. Taylor expanded in c around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

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

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

   8. div-sub N/A

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

   9. sub-neg N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

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

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

   13. /-lowering-/.f32 95.5%

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

8. Simplified 95.5%

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

   1. +-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   12. PI-lowering-PI.f32 96.0%

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

10. Applied egg-rr 96.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{cosTheta} + -1}{{\pi}^{0.5}} + 1}} \]

11. Final simplification 96.0%

    \[\leadsto \frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{{\pi}^{0.5}}} \]
12. Add Preprocessing

Alternative 10: 95.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{\frac{1 - cosTheta}{cosTheta}}{{\pi}^{0.5}}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in c around 0

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

7. Simplified 98.3%

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

8. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\color{blue}{\left(\frac{1 + -1 \cdot cosTheta}{cosTheta}\right)}, \mathsf{pow.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]
9. Step-by-step derivation

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f32 96.0%

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

10. Simplified 96.0%

    \[\leadsto \frac{1}{1 + \frac{\color{blue}{\frac{1 - cosTheta}{cosTheta}}}{{\pi}^{0.5}}} \]
11. Add Preprocessing

Alternative 11: 95.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 95.5%

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

5. Simplified 95.5%

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

6. Taylor expanded in c around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

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

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

   8. div-sub N/A

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

   9. sub-neg N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

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

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

   13. /-lowering-/.f32 95.5%

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

8. Simplified 95.5%

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

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

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

   2. inv-pow N/A

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

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

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

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

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

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

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

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

   8. /-lowering-/.f32 95.5%

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

10. Applied egg-rr 95.5%

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

11. Final simplification 95.5%

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

Alternative 12: 93.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\sqrt{1 - cosTheta \cdot 2}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)}} \]
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 93.6%

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

7. Simplified 93.6%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]
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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in cosTheta around inf

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

10. Simplified 10.7%

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

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. unsub-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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. exp-neg N/A

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in cosTheta around inf

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

10. Simplified 10.7%

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

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 2024161 
(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))))))