Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.4%

Time: 15.3s

Alternatives: 15

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ N/A

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

   2. clear-num N/A

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

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

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

   4. *-lft-identity N/A

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

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

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

   6. pow1/2 N/A

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

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

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

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

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

   9. associate--l- N/A

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

   10. count-2 N/A

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

   11. *-commutative N/A

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

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

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. 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 98.2%

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

6. Applied egg-rr 97.8%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

   \[\leadsto \frac{\frac{1}{\frac{c + -1}{e^{cosTheta \cdot cosTheta} \cdot {\left(\frac{1 + cosTheta \cdot -2}{\pi}\right)}^{-0.5}} + cosTheta \cdot \left(-1 + c \cdot c\right)}}{\frac{\frac{1}{cosTheta}}{c + -1}} \]
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 97.8%

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

   1. /-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 98.2%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\sqrt{1 - cosTheta \cdot 2}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)}} \]
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: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*l/ N/A

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

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

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

5. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{e^{0 - cosTheta \cdot cosTheta} \cdot \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}}{cosTheta}}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{0 - cosTheta \cdot cosTheta} \cdot \color{blue}{\frac{\sqrt{\frac{1 + cosTheta \cdot -2}{\mathsf{PI}\left(\right)}}}{cosTheta}}\right)\right)\right) \]

   2. exp-diff N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(\frac{e^{0}}{e^{cosTheta \cdot cosTheta}} \cdot \frac{\color{blue}{\sqrt{\frac{1 + cosTheta \cdot -2}{\mathsf{PI}\left(\right)}}}}{cosTheta}\right)\right)\right) \]

   3. 1-exp N/A

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

   4. sqrt-div N/A

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

   5. unpow1/2 N/A

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

   6. unpow1/2 N/A

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

   7. clear-num N/A

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

   8. associate-/r* N/A

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

   9. associate-/r/ N/A

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

   10. associate-*l/ N/A

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

   11. div-inv N/A

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

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

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

7. Applied egg-rr 97.5%

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

Alternative 5: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. /-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 98.2%

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

5. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

7. Simplified 97.5%

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

Alternative 6: 97.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. /-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 98.2%

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

6. Applied egg-rr 97.8%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in c around 0

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

    1. associate-*r/ N/A

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

    2. distribute-lft-out N/A

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

    3. times-frac N/A

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

    4. metadata-eval N/A

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

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

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

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

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

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

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

    8. associate-*r/ N/A

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

    9. *-rgt-identity N/A

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

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

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 7: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. /-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 98.2%

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

5. Taylor expanded in cosTheta around 0

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

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

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

7. Simplified 95.8%

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

   1. flip-+ N/A

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

   2. associate-/l/ N/A

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

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

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

9. Applied egg-rr 96.5%

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

10. Final simplification 96.5%

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

Alternative 8: 95.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. /-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 98.2%

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

5. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 96.5%

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

10. Simplified 96.5%

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

11. Final simplification 96.5%

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

Alternative 9: 95.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. 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 98.2%

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

5. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. --lowering--.f32 96.3%

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

10. Simplified 96.3%

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

Alternative 10: 95.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 97.8%

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

   1. /-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 98.2%

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

5. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. PI-lowering-PI.f32 96.0%

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

10. Simplified 96.0%

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

Alternative 11: 94.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. 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 95.8%

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

5. Simplified 95.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. pow1/2 N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

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

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

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

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. --lowering--.f32 95.9%

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

7. Applied egg-rr 95.9%

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

8. Taylor expanded in c around 0

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 12: 92.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. /-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 98.2%

   \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\sqrt{1 - cosTheta \cdot 2}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)}} \]
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 93.0%

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

7. Simplified 93.0%

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

Alternative 13: 10.8% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. 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 98.2%

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

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

10. Simplified 10.6%

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

11. Taylor expanded in c around 0

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

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

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

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

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

    3. sub-neg N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(c, \left(c \cdot \left(1 + -1 \cdot c\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

    4. metadata-eval N/A

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

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

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

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

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

    7. mul-1-neg N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(c, \mathsf{+.f32}\left(\mathsf{*.f32}\left(c, \left(1 + \left(\mathsf{neg}\left(c\right)\right)\right)\right), -1\right)\right)\right) \]

    8. unsub-neg N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(c, \mathsf{+.f32}\left(\mathsf{*.f32}\left(c, \left(1 - c\right)\right), -1\right)\right)\right) \]

    9. --lowering--.f32 10.6%

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(c, \mathsf{+.f32}\left(\mathsf{*.f32}\left(c, \mathsf{\_.f32}\left(1, c\right)\right), -1\right)\right)\right) \]

13. Simplified 10.6%

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

14. Final simplification 10.6%

    \[\leadsto 1 + c \cdot \left(-1 + c \cdot \left(1 - c\right)\right) \]
15. Add Preprocessing

Alternative 14: 10.8% accurate, 107.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. 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 98.2%

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

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

10. Simplified 10.6%

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

11. Taylor expanded in c around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. --lowering--.f32 10.6%

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

13. Simplified 10.6%

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

Alternative 15: 10.8% accurate, 322.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. 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 98.2%

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

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

10. Simplified 10.6%

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

11. Taylor expanded in c around 0

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

13. Simplified 10.6%

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

Reproduce

herbie shell --seed 2024158 
(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))))))