Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 11.5s

Alternatives: 11

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. rem-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.0

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied rewrites 98.3%

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

Alternative 2: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. rem-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.0

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied rewrites 98.3%

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

7. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

9. Applied rewrites 98.0%

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

10. Final simplification 98.0%

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

Alternative 3: 98.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. rem-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.0

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied rewrites 98.3%

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

7. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower-PI.f32 97.7

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

9. Applied rewrites 97.7%

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

Alternative 4: 98.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (/
    (*
     (sqrt (fma cosTheta -2.0 1.0))
     (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(fmaf(cosTheta, -2.0f, 1.0f)) * 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(fma(cosTheta, Float32(-2.0), Float32(1.0))) * 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 97.6%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. rem-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.0

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l/ N/A

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 97.7

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

9. Applied rewrites 97.7%

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

Alternative 5: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 97.7%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 97.1%

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

Alternative 6: 97.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. rem-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.0

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied rewrites 98.3%

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

7. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-PI.f32 97.1

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

9. Applied rewrites 97.1%

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

10. Final simplification 97.1%

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

Alternative 7: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 97.7%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 96.4%

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

Alternative 8: 96.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. lift-sqrt.f32 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. rem-cube-cbrt N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 98.0

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied rewrites 98.3%

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

7. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. exp-neg N/A

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

   8. neg-mul-1 N/A

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

   9. lower-fma.f32 N/A

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

9. Applied rewrites 97.4%

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

10. Taylor expanded in cosTheta around 0

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

12. Applied rewrites 96.2%

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

Alternative 9: 95.9% 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 97.6%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   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. lower-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. Applied rewrites 95.2%

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

Alternative 10: 93.1% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-PI.f32 92.9

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

5. Applied rewrites 92.9%

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

Alternative 11: 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 97.6%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f32 4.9

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

5. Applied rewrites 4.9%

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

Reproduce

herbie shell --seed 2024237 
(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))))))