Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 13.8s

Alternatives: 14

Speedup: 2.4×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f32 98.6

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. associate-*l/ N/A

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

   5. flip3-+ N/A

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

   6. clear-num N/A

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

   7. frac-add N/A

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

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

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 98.2

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

   \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{cosTheta}, \frac{-1}{-1 - c}, \sqrt{\pi}\right)}{\sqrt{\pi} \cdot \frac{-1}{c + 1}}} \]
9. Add Preprocessing

Alternative 3: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f32 98.6

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 98.0

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 4: 98.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 97.5

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

7. Simplified 97.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, -1\right), 1\right)}{cosTheta}}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)} \]
8. Step-by-step derivation

   1. div-sub N/A

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

   2. frac-sub N/A

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

   3. sqrt-div N/A

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

   4. sqrt-unprod N/A

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

   5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 97.4

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

9. Applied egg-rr 97.4%

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

10. Taylor expanded in c around 0

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

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

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

    2. +-commutative N/A

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

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

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

12. Simplified 98.0%

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

13. Final simplification 98.0%

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

Alternative 5: 97.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 97.5

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

7. Simplified 97.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, -1\right), 1\right)}{cosTheta}}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)} \]
8. Step-by-step derivation

   1. div-sub N/A

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

   2. frac-sub N/A

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

   3. sqrt-div N/A

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

   4. sqrt-unprod N/A

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

   5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 97.4

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

9. Applied egg-rr 97.4%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

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

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

12. Simplified 97.5%

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

13. Final simplification 97.5%

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

Alternative 6: 97.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f32 98.6

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unpow2 N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 97.3

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 7: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

   9. neg-lowering-neg.f32 96.8

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 8: 95.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. associate-*l/ N/A

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

   5. flip3-+ N/A

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

   6. clear-num N/A

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

   7. frac-add N/A

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

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

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

4. Applied egg-rr 98.5%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 98.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in cosTheta around 0

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

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

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. *-commutative N/A

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

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

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

    6. mul-1-neg N/A

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

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

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

11. Simplified 95.5%

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

12. Final simplification 95.5%

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

Alternative 9: 95.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f32 98.6

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta around 0

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

7. Simplified 95.6%

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

8. Final simplification 95.6%

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

Alternative 10: 95.7% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

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

Alternative 11: 95.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-out-- N/A

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

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

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

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

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

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

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

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

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

   11. --lowering--.f32 94.7

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

5. Simplified 94.7%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

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

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

7. Applied egg-rr 95.4%

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

8. Final simplification 95.4%

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

Alternative 12: 92.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 94.7

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

5. Simplified 94.7%

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

7. Applied egg-rr 95.9%

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

8. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

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

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 95.5

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

10. Simplified 95.5%

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

11. Taylor expanded in cosTheta around 0

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

    1. mul-1-neg N/A

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

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

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

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

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

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

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

    5. PI-lowering-PI.f32 92.6

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

13. Simplified 92.6%

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

14. Final simplification 92.6%

    \[\leadsto \left(cosTheta \cdot \pi\right) \cdot \sqrt{\frac{1}{\pi}} \]
15. Add Preprocessing

Alternative 13: 92.9% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 92.5

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

5. Simplified 92.5%

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

Alternative 14: 5.1% 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 98.0%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f32 5.3

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

5. Simplified 5.3%

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

Reproduce

herbie shell --seed 2024201 
(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))))))