Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 14.0s

Alternatives: 17

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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\pi}}, \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{{\pi}^{0.16666666666666666}}, 1 + c\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(Float32(Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta) / cbrt(Float32(pi))), Float32(exp(Float32(cosTheta * Float32(-cosTheta))) / (Float32(pi) ^ Float32(0.16666666666666666))), 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

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

4. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\pi}}, \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{{\pi}^{0.16666666666666666}}, 1 + c\right)}} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. frac-times N/A

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

   2. *-lft-identity N/A

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

   3. /-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 98.4

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. frac-times N/A

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

   2. *-lft-identity N/A

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

   3. /-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 98.4

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in c around 0

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

7. Simplified 98.1%

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

8. Final simplification 98.1%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

5. Simplified 98.0%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(exp(Float32(cosTheta * Float32(-cosTheta))), Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) / 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. Taylor expanded in c around 0

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

5. Simplified 97.8%

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

Alternative 6: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 97.7

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

8. Simplified 97.7%

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

Alternative 7: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 97.2

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

8. Simplified 97.2%

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

Alternative 8: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((fmaf(cosTheta, fmaf(cosTheta, fmaf(cosTheta, 0.5f, -1.5f), -1.0f), 1.0f) / cosTheta) / sqrtf(((float) M_PI))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(fma(cosTheta, fma(cosTheta, fma(cosTheta, Float32(0.5), Float32(-1.5)), Float32(-1.0)), Float32(1.0)) / cosTheta) / sqrt(Float32(pi)))))
end

Derivation

1. Initial program 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

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

   2. div-inv N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f32 96.7

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

7. Simplified 96.7%

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

Alternative 9: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

   9. neg-lowering-neg.f32 96.3

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

8. Simplified 96.3%

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

   1. +-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

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

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

10. Applied egg-rr 96.3%

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

Alternative 10: 96.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

   9. neg-lowering-neg.f32 96.3

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

8. Simplified 96.3%

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

Alternative 11: 96.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

   9. neg-lowering-neg.f32 96.3

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

8. Simplified 96.3%

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

9. Taylor expanded in c around 0

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

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

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

    2. +-commutative N/A

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

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

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

11. Simplified 96.1%

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

Alternative 12: 96.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in cosTheta around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f32 95.9

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

7. Simplified 95.9%

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

Alternative 13: 95.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return fma(fma(Float32(pi), Float32(c + Float32(Float32(-1.0) / sqrt(Float32(pi)))), Float32(pi)), Float32(cosTheta * Float32(-cosTheta)), 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 94.7%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

7. Applied egg-rr 94.7%

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

Alternative 14: 95.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(cosTheta * fma(fma(Float32(pi), c, Float32(Float32(pi) - sqrt(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 94.7%

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

6. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

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

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

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

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

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

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

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

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

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

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

   13. PI-lowering-PI.f32 94.7

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

8. Simplified 94.7%

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

Alternative 15: 95.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(cosTheta * fma(Float32(Float32(pi) - sqrt(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 94.7%

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

6. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

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

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

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

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

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

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

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

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

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

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

   14. mul-1-neg N/A

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

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

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

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

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

   17. PI-lowering-PI.f32 94.6

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

8. Simplified 94.6%

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

Alternative 16: 92.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 92.0

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

5. Simplified 92.0%

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

Alternative 17: 5.0% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f32 5.1

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

5. Simplified 5.1%

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

Reproduce

herbie shell --seed 2024204 
(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))))))