Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 12.7s

Alternatives: 12

Speedup: 2.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. frac-times N/A

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

   2. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 98.7

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

4. Applied egg-rr 98.7%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. sqrt-div N/A

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

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

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

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

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

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

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

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

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

   16. --lowering--.f32 N/A

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

   17. --lowering--.f32 98.7

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

6. Applied egg-rr 98.7%

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

Alternative 2: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(Float32(1.0) / fma((Float32(exp(1)) ^ Float32(cosTheta * Float32(-cosTheta))), sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))), cosTheta)))
end

Derivation

1. Initial program 97.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. frac-add N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

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

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

   10. *-commutative N/A

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

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

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

7. Simplified 97.9%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

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

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

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

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

   7. *-lft-identity N/A

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

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

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

9. Applied egg-rr 98.6%

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

    1. *-rgt-identity N/A

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

    2. *-commutative N/A

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

    3. exp-prod N/A

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

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

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

    5. exp-1-e N/A

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

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

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

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

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

    8. neg-lowering-neg.f32 98.6

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

11. Applied egg-rr 98.6%

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

Alternative 3: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. frac-add N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

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

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

   10. *-commutative N/A

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

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

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

7. Simplified 97.9%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

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

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

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

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

   7. *-lft-identity N/A

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

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

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

9. Applied egg-rr 98.6%

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

Alternative 4: 98.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, -0.16666666666666666, 0.5\right), -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, cosTheta\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (*
  cosTheta
  (/
   1.0
   (fma
    (fma
     (* cosTheta cosTheta)
     (fma
      (* cosTheta cosTheta)
      (fma (* cosTheta cosTheta) -0.16666666666666666 0.5)
      -1.0)
     1.0)
    (sqrt (/ (fma cosTheta -2.0 1.0) PI))
    cosTheta))))

C

float code(float cosTheta, float c) {
	return cosTheta * (1.0f / fmaf(fmaf((cosTheta * cosTheta), fmaf((cosTheta * cosTheta), fmaf((cosTheta * cosTheta), -0.16666666666666666f, 0.5f), -1.0f), 1.0f), sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI))), cosTheta));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(Float32(1.0) / fma(fma(Float32(cosTheta * cosTheta), fma(Float32(cosTheta * cosTheta), fma(Float32(cosTheta * cosTheta), Float32(-0.16666666666666666), Float32(0.5)), Float32(-1.0)), Float32(1.0)), sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))), cosTheta)))
end

Derivation

1. Initial program 97.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. frac-add N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

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

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

   10. *-commutative N/A

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

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

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

7. Simplified 97.9%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

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

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

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

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

   7. *-lft-identity N/A

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

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

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in cosTheta around 0

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

    1. +-commutative N/A

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\color{blue}{{cosTheta}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1\right) + 1}, \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

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

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({cosTheta}^{2}, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, 1\right)}, \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    3. unpow2 N/A

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

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

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    5. sub-neg N/A

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{{cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    6. metadata-eval N/A

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) + \color{blue}{-1}, 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

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

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{\mathsf{fma}\left({cosTheta}^{2}, \frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}, -1\right)}, 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    8. unpow2 N/A

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, \frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

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

       \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, \frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    10. +-commutative N/A

        \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{\frac{-1}{6} \cdot {cosTheta}^{2} + \frac{1}{2}}, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    11. *-commutative N/A

        \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{{cosTheta}^{2} \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

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

        \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{\mathsf{fma}\left({cosTheta}^{2}, \frac{-1}{6}, \frac{1}{2}\right)}, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    13. unpow2 N/A

        \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    14. *-lowering-*.f32 98.6

        \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, -0.16666666666666666, 0.5\right), -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, cosTheta\right)} \]

12. Simplified 98.6%

    \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, -0.16666666666666666, 0.5\right), -1\right), 1\right)}, \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, cosTheta\right)} \]
13. Add Preprocessing

Alternative 5: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, cosTheta\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (*
  cosTheta
  (/
   1.0
   (fma
    (fma (* cosTheta cosTheta) (fma (* cosTheta cosTheta) 0.5 -1.0) 1.0)
    (sqrt (/ (fma cosTheta -2.0 1.0) PI))
    cosTheta))))

C

float code(float cosTheta, float c) {
	return cosTheta * (1.0f / fmaf(fmaf((cosTheta * cosTheta), fmaf((cosTheta * cosTheta), 0.5f, -1.0f), 1.0f), sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI))), cosTheta));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(Float32(1.0) / fma(fma(Float32(cosTheta * cosTheta), fma(Float32(cosTheta * cosTheta), Float32(0.5), Float32(-1.0)), Float32(1.0)), sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))), cosTheta)))
end

Derivation

1. Initial program 97.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. frac-add N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

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

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

   10. *-commutative N/A

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

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

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

7. Simplified 97.9%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

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

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

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

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

   7. *-lft-identity N/A

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

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

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in cosTheta around 0

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

    1. +-commutative N/A

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

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

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

    3. unpow2 N/A

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

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

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

    5. sub-neg N/A

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

    6. *-commutative N/A

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

    7. metadata-eval N/A

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f32 98.5

        \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, 0.5, -1\right), 1\right), \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, cosTheta\right)} \]

12. Simplified 98.5%

    \[\leadsto cosTheta \cdot \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, -1\right), 1\right)}, \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, cosTheta\right)} \]
13. Add Preprocessing

Alternative 6: 97.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. frac-add N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

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

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

   10. *-commutative N/A

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

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

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

7. Simplified 97.9%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

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

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

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

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

   7. *-lft-identity N/A

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

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

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in cosTheta around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unpow2 N/A

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

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

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

    5. mul-1-neg N/A

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

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

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

    7. mul-1-neg N/A

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

    8. neg-lowering-neg.f32 97.9

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

12. Simplified 97.9%

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

Alternative 7: 95.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. frac-add N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

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

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

   10. *-commutative N/A

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

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

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

7. Simplified 97.9%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

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

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

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

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

   7. *-lft-identity N/A

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

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

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in cosTheta around 0

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

12. Simplified 96.3%

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

Alternative 8: 95.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 96.1%

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

6. Taylor expanded in c around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. mul-1-neg N/A

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

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

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

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

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

   10. PI-lowering-PI.f32 96.1

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

8. Simplified 96.1%

   \[\leadsto cosTheta \cdot \mathsf{fma}\left(\color{blue}{\pi + \mathsf{fma}\left(\pi, c, -\sqrt{\pi}\right)}, -cosTheta, \sqrt{\pi}\right) \]
9. Add Preprocessing

Alternative 9: 95.4% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 96.1%

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

6. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 96.1

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

8. Simplified 96.1%

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

Alternative 10: 92.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 96.1%

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

6. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

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

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

   3. PI-lowering-PI.f32 94.0

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

8. Simplified 94.0%

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

9. Final simplification 94.0%

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

Alternative 11: 92.7% 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 97.6%

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

3. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 94.0

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

5. Simplified 94.0%

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

Alternative 12: 5.0% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f32 4.7

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

5. Simplified 4.7%

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

Reproduce

herbie shell --seed 2024199 
(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))))))