Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.2%

Time: 20.1s

Alternatives: 14

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. +-commutative N/A

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

   5. lift-neg.f32 N/A

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

   6. neg-sub0 N/A

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

   7. flip-- N/A

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

   8. metadata-eval N/A

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

   9. neg-sub0 N/A

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

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

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

   11. lift-neg.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. flip3-- N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\frac{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}{0 + cosTheta} + \color{blue}{\frac{{1}^{3} - {cosTheta}^{3}}{1 \cdot 1 + \left(cosTheta \cdot cosTheta + 1 \cdot cosTheta\right)}}}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}} \]

   15. clear-num N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\frac{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}{0 + cosTheta} + \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(cosTheta \cdot cosTheta + 1 \cdot cosTheta\right)}{{1}^{3} - {cosTheta}^{3}}}}}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}} \]

   16. frac-add N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\color{blue}{\frac{\left(\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta\right) \cdot \frac{1 \cdot 1 + \left(cosTheta \cdot cosTheta + 1 \cdot cosTheta\right)}{{1}^{3} - {cosTheta}^{3}} + \left(0 + cosTheta\right) \cdot 1}{\left(0 + cosTheta\right) \cdot \frac{1 \cdot 1 + \left(cosTheta \cdot cosTheta + 1 \cdot cosTheta\right)}{{1}^{3} - {cosTheta}^{3}}}}}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}} \]

   17. clear-num N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\frac{\left(\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta\right) \cdot \frac{1 \cdot 1 + \left(cosTheta \cdot cosTheta + 1 \cdot cosTheta\right)}{{1}^{3} - {cosTheta}^{3}} + \left(0 + cosTheta\right) \cdot 1}{\left(0 + cosTheta\right) \cdot \color{blue}{\frac{1}{\frac{{1}^{3} - {cosTheta}^{3}}{1 \cdot 1 + \left(cosTheta \cdot cosTheta + 1 \cdot cosTheta\right)}}}}}}{cosTheta}\right) \cdot e^{\left(\mathsf{neg}\left(cosTheta\right)\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.5%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. div-inv N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-+.f32 97.4

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

4. Applied egg-rr 97.4%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. associate-*l/ N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f32 N/A

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

7. Applied egg-rr 97.8%

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

8. Final simplification 97.8%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lower-+.f32 N/A

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

7. Applied egg-rr 97.7%

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

Alternative 5: 96.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := \left(c + 1\right) - t\_0\\ cosTheta \cdot \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\pi, t\_0 \cdot 1.5, t\_1 \cdot \left(t\_1 \cdot \sqrt{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right), \sqrt{\pi} - \pi\right), \sqrt{\pi}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (- (+ c 1.0) t_0)))
   (*
    cosTheta
    (fma
     cosTheta
     (fma
      cosTheta
      (fma PI (* t_0 1.5) (* t_1 (* t_1 (sqrt (* PI (* PI PI))))))
      (- (sqrt PI) PI))
     (sqrt PI)))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	float t_1 = (c + 1.0f) - t_0;
	return cosTheta * fmaf(cosTheta, fmaf(cosTheta, fmaf(((float) M_PI), (t_0 * 1.5f), (t_1 * (t_1 * sqrtf((((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))), (sqrtf(((float) M_PI)) - ((float) M_PI))), sqrtf(((float) M_PI)));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	t_1 = Float32(Float32(c + Float32(1.0)) - t_0)
	return Float32(cosTheta * fma(cosTheta, fma(cosTheta, fma(Float32(pi), Float32(t_0 * Float32(1.5)), Float32(t_1 * Float32(t_1 * sqrt(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))), Float32(sqrt(Float32(pi)) - Float32(pi))), sqrt(Float32(pi))))
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

5. Simplified 97.5%

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

6. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-PI.f32 97.4

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 6: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(t\_0, -1.5, t\_0 \cdot \left(cosTheta \cdot 0.5\right)\right), 1 - t\_0\right), t\_0\right)}{cosTheta}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (/
     (fma
      cosTheta
      (fma cosTheta (fma t_0 -1.5 (* t_0 (* cosTheta 0.5))) (- 1.0 t_0))
      t_0)
     cosTheta))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (fmaf(cosTheta, fmaf(cosTheta, fmaf(t_0, -1.5f, (t_0 * (cosTheta * 0.5f))), (1.0f - t_0)), t_0) / cosTheta);
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(fma(cosTheta, fma(cosTheta, fma(t_0, Float32(-1.5), Float32(t_0 * Float32(cosTheta * Float32(0.5)))), Float32(Float32(1.0) - t_0)), t_0) / cosTheta))
end

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\sqrt{\frac{1}{\pi}}, -1.5, \sqrt{\frac{1}{\pi}} \cdot \left(cosTheta \cdot 0.5\right)\right), 1 - \sqrt{\frac{1}{\pi}}\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}} \]
10. Add Preprocessing

Alternative 7: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, t\_0 \cdot \left(cosTheta + -1.5\right), 1 - t\_0\right), t\_0\right)}{cosTheta}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (/
     (fma cosTheta (fma cosTheta (* t_0 (+ cosTheta -1.5)) (- 1.0 t_0)) t_0)
     cosTheta))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (fmaf(cosTheta, fmaf(cosTheta, (t_0 * (cosTheta + -1.5f)), (1.0f - t_0)), t_0) / cosTheta);
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(fma(cosTheta, fma(cosTheta, Float32(t_0 * Float32(cosTheta + Float32(-1.5))), Float32(Float32(1.0) - t_0)), t_0) / cosTheta))
end

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

8. Simplified 96.5%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-/.f32 N/A

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

11. Simplified 96.7%

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

12. Final simplification 96.7%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \left(cosTheta + -1.5\right), 1 - \sqrt{\frac{1}{\pi}}\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}} \]
13. Add Preprocessing

Alternative 8: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(t\_0, cosTheta \cdot -1.5, 1 - t\_0\right), t\_0\right)}{cosTheta}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (/ (fma cosTheta (fma t_0 (* cosTheta -1.5) (- 1.0 t_0)) t_0) cosTheta))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (fmaf(cosTheta, fmaf(t_0, (cosTheta * -1.5f), (1.0f - t_0)), t_0) / cosTheta);
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(fma(cosTheta, fma(t_0, Float32(cosTheta * Float32(-1.5)), Float32(Float32(1.0) - t_0)), t_0) / cosTheta))
end

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\sqrt{\frac{1}{\pi}}, cosTheta \cdot -1.5, 1 - \sqrt{\frac{1}{\pi}}\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}} \]
10. Add Preprocessing

Alternative 9: 96.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi - \sqrt{\pi}\\ \frac{cosTheta \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, t\_0 \cdot t\_0, -\pi\right)}{\mathsf{fma}\left(-cosTheta, t\_0, -\sqrt{\pi}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (- PI (sqrt PI))))
   (/
    (* cosTheta (fma (* cosTheta cosTheta) (* t_0 t_0) (- PI)))
    (fma (- cosTheta) t_0 (- (sqrt PI))))))

C

float code(float cosTheta, float c) {
	float t_0 = ((float) M_PI) - sqrtf(((float) M_PI));
	return (cosTheta * fmaf((cosTheta * cosTheta), (t_0 * t_0), -((float) M_PI))) / fmaf(-cosTheta, t_0, -sqrtf(((float) M_PI)));
}

Julia

function code(cosTheta, c)
	t_0 = Float32(Float32(pi) - sqrt(Float32(pi)))
	return Float32(Float32(cosTheta * fma(Float32(cosTheta * cosTheta), Float32(t_0 * t_0), Float32(-Float32(pi)))) / fma(Float32(-cosTheta), t_0, Float32(-sqrt(Float32(pi)))))
end

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. neg-mul-1 N/A

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

   7. lower-fma.f32 N/A

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

8. Simplified 96.4%

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

10. Applied egg-rr 96.6%

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

11. Final simplification 96.6%

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

Alternative 10: 95.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. neg-mul-1 N/A

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

   7. lower-fma.f32 N/A

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

8. Simplified 96.4%

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

10. Applied egg-rr 96.4%

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

11. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\sqrt{\pi}, cosTheta, cosTheta \cdot \left(cosTheta \cdot \left(\sqrt{\pi} - \pi\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 95.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Simplified 97.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. neg-mul-1 N/A

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

   7. lower-fma.f32 N/A

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

8. Simplified 96.4%

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

10. Applied egg-rr 96.4%

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

11. Final simplification 96.4%

    \[\leadsto cosTheta \cdot \mathsf{fma}\left(-cosTheta, \pi - \sqrt{\pi}, \sqrt{\pi}\right) \]
12. Add Preprocessing

Alternative 12: 92.9% accurate, 11.4× 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 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-PI.f32 94.1

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

5. Simplified 94.1%

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

Alternative 13: 10.8% accurate, 45.8× 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 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 93.0

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

5. Simplified 93.0%

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

6. Taylor expanded in cosTheta around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 10.5

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

8. Simplified 10.5%

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

9. Taylor expanded in c around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 10.5

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

11. Simplified 10.5%

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

Alternative 14: 10.8% accurate, 183.0× 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 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 93.0

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

5. Simplified 93.0%

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

6. Taylor expanded in cosTheta around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 10.5

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

8. Simplified 10.5%

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

9. Taylor expanded in c around 0

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

11. Simplified 10.4%

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

Reproduce

herbie shell --seed 2024219 
(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))))))