Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.0%

Time: 12.8s

Alternatives: 13

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% 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: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

6. Taylor expanded in c around 0

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

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. lower-fma.f32 N/A

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

8. Simplified 98.1%

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

Alternative 2: 97.8% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/
    1.0
    (fma
     cosTheta
     (*
      (* cosTheta cosTheta)
      (fma
       (* cosTheta cosTheta)
       (fma (* cosTheta cosTheta) 0.16666666666666666 0.5)
       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((1.0f / fmaf(cosTheta, ((cosTheta * cosTheta) * fmaf((cosTheta * cosTheta), fmaf((cosTheta * cosTheta), 0.16666666666666666f, 0.5f), 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(Float32(1.0) / fma(cosTheta, Float32(Float32(cosTheta * cosTheta) * fma(Float32(cosTheta * cosTheta), fma(Float32(cosTheta * cosTheta), Float32(0.16666666666666666), Float32(0.5)), 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 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

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

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. div-inv N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-exp.f32 98.1

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

7. Applied egg-rr 98.1%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 98.0

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

10. Simplified 98.0%

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

Alternative 3: 97.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/
    (fma
     (* cosTheta cosTheta)
     (fma
      (* cosTheta cosTheta)
      (fma (* cosTheta cosTheta) -0.16666666666666666 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), fmaf((cosTheta * cosTheta), -0.16666666666666666f, 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), fma(Float32(cosTheta * cosTheta), Float32(-0.16666666666666666), 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 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 98.0

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

8. Simplified 98.0%

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

Alternative 4: 97.5% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/
    1.0
    (fma
     cosTheta
     (* (* cosTheta cosTheta) (fma (* cosTheta cosTheta) 0.5 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((1.0f / fmaf(cosTheta, ((cosTheta * cosTheta) * fmaf((cosTheta * cosTheta), 0.5f, 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(Float32(1.0) / fma(cosTheta, Float32(Float32(cosTheta * cosTheta) * fma(Float32(cosTheta * cosTheta), Float32(0.5), 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 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

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

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. div-inv N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-exp.f32 98.1

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

7. Applied egg-rr 98.1%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 97.8

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

10. Simplified 97.8%

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

Alternative 5: 97.6% 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 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 97.8

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

8. Simplified 97.8%

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

Alternative 6: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

6. Taylor expanded in c around 0

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

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. lower-fma.f32 N/A

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

8. Simplified 98.1%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f32 N/A

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

    6. sub-neg N/A

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

    7. *-commutative N/A

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

    8. metadata-eval N/A

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

    9. lower-fma.f32 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f32 97.8

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

11. Simplified 97.8%

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

Alternative 7: 96.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf((1.0f / fmaf(cosTheta, (cosTheta * cosTheta), 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(Float32(1.0) / fma(cosTheta, Float32(cosTheta * cosTheta), cosTheta)), sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))), Float32(Float32(1.0) + c)))
end

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

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

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. div-inv N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-exp.f32 98.1

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

7. Applied egg-rr 98.1%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 97.3

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

10. Simplified 97.3%

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

Alternative 8: 96.9% 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 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. neg-mul-1 N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f32 97.3

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

8. Simplified 97.3%

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

Alternative 9: 95.6% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in cosTheta around 0

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

   1. 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. Simplified 96.9%

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

Alternative 10: 95.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 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. Simplified 98.1%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 96.3

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

8. Simplified 96.3%

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

9. Taylor expanded in c around 0

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

    1. lower-/.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. associate-*l/ N/A

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

    4. *-lft-identity N/A

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

    5. lower-/.f32 N/A

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

    6. lower-sqrt.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. +-commutative N/A

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

    9. *-commutative N/A

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

    10. rem-square-sqrt N/A

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

    11. unpow2 N/A

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

    12. lower-fma.f32 N/A

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

    13. unpow2 N/A

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

    14. rem-square-sqrt N/A

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

    15. lower-PI.f32 96.2

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

11. Simplified 96.2%

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

Alternative 11: 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 98.0%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-PI.f32 94.0

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

5. Simplified 94.0%

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

Alternative 12: 10.8% accurate, 45.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

4. Applied egg-rr 93.4%

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

6. Applied egg-rr 93.6%

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

7. Taylor expanded in cosTheta around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 10.5

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

9. Simplified 10.5%

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

10. Taylor expanded in c around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f32 10.5

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

12. Simplified 10.5%

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

Alternative 13: 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 98.0%

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

4. Applied egg-rr 93.4%

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

6. Applied egg-rr 93.6%

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

7. Taylor expanded in cosTheta around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 10.5

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

9. Simplified 10.5%

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

10. Taylor expanded in c around 0

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

12. Simplified 10.5%

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

Reproduce

herbie shell --seed 2024207 
(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))))))