Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.4%

Time: 13.4s

Alternatives: 10

Speedup: 2.0×

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: 98.4% accurate, 1.0× speedup

Math

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

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(Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) * exp(Float32(cosTheta * Float32(-cosTheta)))) / 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 97.7%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

4. Applied egg-rr 98.2%

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

Alternative 2: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c + -1, 1\right)\\ \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot c, 1\right), cosTheta, t\_0 \cdot \left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, -0.16666666666666666, 0.5\right), -1\right), 1\right)\right)\right)}{cosTheta \cdot t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (fma c (+ c -1.0) 1.0)))
   (/
    1.0
    (/
     (fma
      (fma c (* c c) 1.0)
      cosTheta
      (*
       t_0
       (*
        (sqrt (/ (- (- 1.0 cosTheta) cosTheta) PI))
        (fma
         (* cosTheta cosTheta)
         (fma
          (* cosTheta cosTheta)
          (fma (* cosTheta cosTheta) -0.16666666666666666 0.5)
          -1.0)
         1.0))))
     (* cosTheta t_0)))))

C

float code(float cosTheta, float c) {
	float t_0 = fmaf(c, (c + -1.0f), 1.0f);
	return 1.0f / (fmaf(fmaf(c, (c * c), 1.0f), cosTheta, (t_0 * (sqrtf((((1.0f - cosTheta) - cosTheta) / ((float) M_PI))) * fmaf((cosTheta * cosTheta), fmaf((cosTheta * cosTheta), fmaf((cosTheta * cosTheta), -0.16666666666666666f, 0.5f), -1.0f), 1.0f)))) / (cosTheta * t_0));
}

Julia

function code(cosTheta, c)
	t_0 = fma(c, Float32(c + Float32(-1.0)), Float32(1.0))
	return Float32(Float32(1.0) / Float32(fma(fma(c, Float32(c * c), Float32(1.0)), cosTheta, Float32(t_0 * Float32(sqrt(Float32(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta) / Float32(pi))) * fma(Float32(cosTheta * cosTheta), fma(Float32(cosTheta * cosTheta), fma(Float32(cosTheta * cosTheta), Float32(-0.16666666666666666), Float32(0.5)), Float32(-1.0)), Float32(1.0))))) / Float32(cosTheta * t_0)))
end

Derivation

1. Initial program 97.7%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.8

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 3: 97.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.8

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

7. Simplified 97.8%

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

   1. clear-num N/A

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

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

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

   3. *-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. *-lft-identity N/A

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

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

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

   8. distribute-lft-in N/A

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

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

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

   10. *-commutative N/A

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

   11. mul-1-neg N/A

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

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

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

9. Applied egg-rr 97.8%

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

10. Taylor expanded in c around 0

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

12. Simplified 97.8%

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

13. Final simplification 97.8%

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

Alternative 4: 98.0% accurate, 2.0× speedup

Math

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

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(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))) / cosTheta) / sqrt(Float32(pi)))))
end

Derivation

1. Initial program 97.7%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 97.5

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

7. Simplified 97.5%

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

Alternative 5: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 97.8

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

7. Simplified 97.8%

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

8. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

10. Simplified 97.4%

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

Alternative 6: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \frac{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right), 1\right)}{cosTheta}}{\sqrt{\pi}}} \end{array} \]

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((fmaf(cosTheta, fmaf(cosTheta, fmaf(cosTheta, 0.5f, -1.5f), -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(fma(cosTheta, fma(cosTheta, fma(cosTheta, Float32(0.5), Float32(-1.5)), Float32(-1.0)), Float32(1.0)) / cosTheta) / sqrt(Float32(pi)))))
end

Derivation

1. Initial program 97.7%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f32 96.7

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

7. Simplified 96.7%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right), 1\right)}{cosTheta}}}{\sqrt{\pi}}} \]
8. Add Preprocessing

Alternative 7: 96.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unpow2 N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 96.6

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

7. Simplified 96.6%

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

8. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

10. Simplified 96.5%

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

Alternative 8: 96.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

6. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. +-commutative N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

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

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

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

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

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

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

   13. PI-lowering-PI.f32 95.9

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

8. Simplified 95.9%

   \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, -1.5, -1\right), 1\right), \sqrt{\frac{1}{\pi}}, cosTheta\right)}}{cosTheta}} \]
9. Step-by-step derivation

   1. clear-num N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   12. PI-lowering-PI.f32 95.9

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

10. Applied egg-rr 95.9%

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

Alternative 9: 92.8% 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 97.7%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 91.9

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

5. Simplified 91.9%

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

Alternative 10: 10.8% accurate, 183.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in c around inf

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

5. Simplified 60.4%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 53.6

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

8. Simplified 53.6%

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

9. Taylor expanded in cosTheta around inf

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

11. Simplified 11.0%

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

Reproduce

herbie shell --seed 2024198 
(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))))))