Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.6%

Time: 12.7s

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: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\pi}} \cdot e^{0 - cosTheta \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) (* cosTheta (sqrt PI)))
    (exp (- 0.0 (* cosTheta cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((sqrtf(((1.0f - cosTheta) - cosTheta)) / (cosTheta * sqrtf(((float) M_PI)))) * expf((0.0f - (cosTheta * cosTheta)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / Float32(cosTheta * sqrt(Float32(pi)))) * exp(Float32(Float32(0.0) - Float32(cosTheta * cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((sqrt(((single(1.0) - cosTheta) - cosTheta)) / (cosTheta * sqrt(single(pi)))) * exp((single(0.0) - (cosTheta * 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. Step-by-step derivation

   1. frac-times N/A

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

   2. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 98.5

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\pi}} \cdot e^{0 - cosTheta \cdot cosTheta}} \]
6. Add Preprocessing

Alternative 2: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/ (exp (- 0.0 (* cosTheta cosTheta))) cosTheta)
   (sqrt (/ (- (- 1.0 cosTheta) cosTheta) PI))
   (+ 1.0 c))))

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf((expf((0.0f - (cosTheta * cosTheta))) / cosTheta), sqrtf((((1.0f - cosTheta) - cosTheta) / ((float) M_PI))), (1.0f + c));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(Float32(exp(Float32(Float32(0.0) - Float32(cosTheta * cosTheta))) / cosTheta), sqrt(Float32(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta) / 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. distribute-lft-neg-out N/A

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

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

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

   4. neg-lowering-neg.f32 98.0

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)} \]
8. Add Preprocessing

Alternative 3: 97.8% accurate, 2.0× speedup

Math

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

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 (/ (- (- 1.0 cosTheta) cosTheta) 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((((1.0f - cosTheta) - cosTheta) / ((float) M_PI))), (1.0f + c));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(Float32(fma(Float32(cosTheta * cosTheta), fma(cosTheta, Float32(cosTheta * fma(Float32(cosTheta * cosTheta), Float32(-0.16666666666666666), Float32(0.5))), Float32(-1.0)), Float32(1.0)) / cosTheta), sqrt(Float32(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta) / 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)}} \]

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 97.9

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

7. Simplified 97.9%

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

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/
    (fma (* cosTheta cosTheta) (fma (* cosTheta cosTheta) 0.5 -1.0) 1.0)
    cosTheta)
   (/ (sqrt (- (* (- 1.0 cosTheta) PI) (* cosTheta PI))) 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((((1.0f - cosTheta) * ((float) M_PI)) - (cosTheta * ((float) M_PI)))) / ((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), Float32(sqrt(Float32(Float32(Float32(Float32(1.0) - cosTheta) * Float32(pi)) - Float32(cosTheta * Float32(pi)))) / 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)}} \]

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 97.5

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

7. Simplified 97.5%

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

   1. div-sub N/A

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

   2. frac-sub N/A

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

   3. sqrt-div N/A

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

   4. sqrt-unprod N/A

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

   5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 97.5

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

9. Applied egg-rr 97.5%

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

10. Final simplification 97.5%

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

Alternative 5: 97.5% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/
    (fma (* cosTheta cosTheta) (fma (* cosTheta cosTheta) 0.5 -1.0) 1.0)
    cosTheta)
   (sqrt (/ (- (- 1.0 cosTheta) cosTheta) 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((((1.0f - cosTheta) - cosTheta) / ((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(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta) / 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)}} \]

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 97.5

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

7. Simplified 97.5%

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

Alternative 6: 97.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)}} \]

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 97.5

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

7. Simplified 97.5%

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

8. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

10. Simplified 96.8%

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

Alternative 7: 96.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf((fmaf(cosTheta, -cosTheta, 1.0f) / cosTheta), sqrtf((((1.0f - cosTheta) - cosTheta) / ((float) M_PI))), (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(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta) / 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. un-div-inv N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

4. Applied egg-rr 98.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, 1 + c\right)}} \]

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. neg-mul-1 N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

   9. neg-lowering-neg.f32 96.8

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

7. Simplified 96.8%

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

Alternative 8: 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. *-lowering-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 95.1%

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

Alternative 9: 95.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

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

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

   2. +-commutative N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 94.6

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

5. Simplified 94.6%

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

6. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. PI-lowering-PI.f32 94.3

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

8. Simplified 94.3%

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

9. Taylor expanded in cosTheta around 0

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

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

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. associate-*r* N/A

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

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

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

    6. mul-1-neg N/A

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

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

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

11. Simplified 95.0%

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

Alternative 10: 95.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(-1.0) + (single(1.0) / 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

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

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

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

   2. +-commutative N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 94.6

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

5. Simplified 94.6%

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

6. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. PI-lowering-PI.f32 94.3

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

8. Simplified 94.3%

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

9. Taylor expanded in cosTheta around inf

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

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

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

    2. associate--l+ N/A

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

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

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

    4. sub-neg N/A

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

    5. mul-1-neg N/A

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

    6. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

    12. /-lowering-/.f32 94.1

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

11. Simplified 94.1%

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

    1. associate-/r* N/A

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

    2. *-inverses N/A

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

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

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

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

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

    5. *-commutative N/A

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

    6. sqrt-div N/A

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

    7. metadata-eval N/A

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

    8. un-div-inv N/A

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

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

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

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

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

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

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

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

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

    13. PI-lowering-PI.f32 94.9

        \[\leadsto \frac{1}{1 + \frac{\frac{1}{cosTheta} + -1}{\sqrt{\color{blue}{\pi}}}} \]

13. Applied egg-rr 94.9%

    \[\leadsto \color{blue}{\frac{1}{1 + \frac{\frac{1}{cosTheta} + -1}{\sqrt{\pi}}}} \]

14. Final simplification 94.9%

    \[\leadsto \frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \]
15. Add Preprocessing

Alternative 11: 94.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 94.6

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

5. Simplified 94.6%

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

6. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. PI-lowering-PI.f32 94.3

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

8. Simplified 94.3%

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

9. Taylor expanded in cosTheta around inf

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

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

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

    2. associate--l+ N/A

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

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

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

    4. sub-neg N/A

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

    5. mul-1-neg N/A

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

    6. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

    12. /-lowering-/.f32 94.1

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

11. Simplified 94.1%

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

12. Taylor expanded in cosTheta around 0

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. *-lft-identity N/A

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

    4. associate-+l+ N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. +-commutative N/A

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

    8. associate-*r* N/A

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

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

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

    10. distribute-rgt1-in N/A

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

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

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

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

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

    13. mul-1-neg N/A

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

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

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

14. Simplified 94.3%

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

15. Final simplification 94.3%

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

Alternative 12: 92.7% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 93.2

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

5. Simplified 93.2%

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

Alternative 13: 4.9% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around inf

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

5. Simplified 5.2%

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

Reproduce

herbie shell --seed 2024196 
(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))))))