Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.3%

Time: 12.9s

Alternatives: 12

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

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

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

   2. exp-neg N/A

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

   3. div-inv N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l/ N/A

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

   8. clear-num N/A

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

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

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

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

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

Alternative 3: 98.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 98.1%

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

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

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

   2. exp-neg N/A

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

   3. div-inv N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l/ N/A

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

   8. clear-num N/A

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

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

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

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

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

4. Applied egg-rr 98.5%

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

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. 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

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

7. Simplified 98.2%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f32 98.2%

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

9. Applied egg-rr 98.2%

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

Alternative 4: 98.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

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

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

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

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

7. Simplified 98.2%

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

Alternative 5: 97.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 96.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 7: 95.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f32 96.2%

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

7. Simplified 96.2%

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

   1. clear-num N/A

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

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

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

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

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

   4. pow1/2 N/A

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

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

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

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

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

   7. div-sub N/A

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

   8. sub-neg N/A

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

   9. *-inverses N/A

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

   10. metadata-eval N/A

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

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

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

   12. /-lowering-/.f32 96.3%

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

9. Applied egg-rr 96.3%

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

10. Final simplification 96.3%

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

Alternative 8: 95.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f32 96.2%

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

7. Simplified 96.2%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

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

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

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

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

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

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

   6. div-sub N/A

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

   7. sub-neg N/A

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

   8. *-inverses N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. pow1/2 N/A

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

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

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

   14. PI-lowering-PI.f32 96.2%

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

9. Applied egg-rr 96.2%

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

10. Final simplification 96.2%

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

Alternative 9: 95.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f32 96.2%

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

7. Simplified 96.2%

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

Alternative 10: 95.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. +-lowering-+.f32 N/A

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

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

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

   4. associate-*l* N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

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

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. --lowering--.f32 96.2%

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

7. Simplified 96.2%

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

8. Taylor expanded in c around 0

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

10. Simplified 95.9%

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

Alternative 11: 92.7% 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 98.1%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}} \]
4. 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 93.5%

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

5. Simplified 93.5%

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

Alternative 12: 5.0% accurate, 107.3× speedup

Math

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

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 98.1%

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

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{1}{c}} \]
4. Step-by-step derivation

   1. /-lowering-/.f32 4.8%

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

5. Simplified 4.8%

   \[\leadsto \color{blue}{\frac{1}{c}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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))))))