Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.1%

Time: 14.5s

Alternatives: 15

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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. +-commutative N/A

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

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

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\sqrt{\frac{\color{blue}{1 + -2 \cdot cosTheta}}{\mathsf{PI}\left(\right)}}}{cosTheta}, e^{0 - cosTheta \cdot cosTheta}, 1 + c\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\sqrt{\frac{\color{blue}{-2 \cdot cosTheta + 1}}{\mathsf{PI}\left(\right)}}}{cosTheta}, e^{0 - cosTheta \cdot cosTheta}, 1 + c\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\sqrt{\frac{\color{blue}{cosTheta \cdot -2} + 1}{\mathsf{PI}\left(\right)}}}{cosTheta}, e^{0 - cosTheta \cdot cosTheta}, 1 + c\right)} \]

   3. accelerator-lowering-fma.f32 98.2

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\pi}}}{cosTheta}, e^{0 - cosTheta \cdot cosTheta}, 1 + c\right)} \]

7. Simplified 98.2%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\pi}}}{cosTheta}, e^{0 - cosTheta \cdot cosTheta}, 1 + c\right)} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

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

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

   4. neg-lowering-neg.f32 98.2

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

9. Applied egg-rr 98.2%

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

10. Final simplification 98.2%

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

Alternative 2: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. +-commutative N/A

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

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

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

4. Applied egg-rr 98.2%

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

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

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

   2. +-lft-identity N/A

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

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

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

   4. neg-lowering-neg.f32 98.2

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

6. Applied egg-rr 98.2%

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

7. Final simplification 98.2%

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

Alternative 3: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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);
}

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(1.0)))
end

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 97.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{e^{0 - cosTheta \cdot cosTheta}}{cosTheta}, \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}, 1\right)}} \]

6. Final simplification 97.4%

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

Alternative 4: 97.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\frac{\mathsf{fma}\left(cosTheta, 1 + \mathsf{fma}\left(cosTheta, t\_0 \cdot \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), c - t\_0\right), t\_0\right)}{cosTheta}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (/
     (fma
      cosTheta
      (+ 1.0 (fma cosTheta (* t_0 (fma cosTheta 0.5 -1.5)) (- c t_0)))
      t_0)
     cosTheta))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (fmaf(cosTheta, (1.0f + fmaf(cosTheta, (t_0 * fmaf(cosTheta, 0.5f, -1.5f)), (c - t_0))), t_0) / cosTheta);
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(fma(cosTheta, Float32(Float32(1.0) + fma(cosTheta, Float32(t_0 * fma(cosTheta, Float32(0.5), Float32(-1.5))), Float32(c - t_0))), t_0) / cosTheta))
end

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.6%

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

Alternative 5: 97.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\left(1 + c\right) + \frac{\mathsf{fma}\left(t\_0, 1 - cosTheta, \left(cosTheta \cdot cosTheta\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(cosTheta, 0.5, -1.5\right)\right)\right)}{cosTheta}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (+
     (+ 1.0 c)
     (/
      (fma
       t_0
       (- 1.0 cosTheta)
       (* (* cosTheta cosTheta) (* t_0 (fma cosTheta 0.5 -1.5))))
      cosTheta)))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / ((1.0f + c) + (fmaf(t_0, (1.0f - cosTheta), ((cosTheta * cosTheta) * (t_0 * fmaf(cosTheta, 0.5f, -1.5f)))) / cosTheta));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(fma(t_0, Float32(Float32(1.0) - cosTheta), Float32(Float32(cosTheta * cosTheta) * Float32(t_0 * fma(cosTheta, Float32(0.5), Float32(-1.5))))) / cosTheta)))
end

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.6%

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

Alternative 6: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

   1. div-inv N/A

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

   2. associate-/r* N/A

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

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

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

7. Applied egg-rr 96.2%

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

Alternative 7: 96.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{\left(c + \frac{t\_0}{cosTheta}\right) + \mathsf{fma}\left(t\_0, \mathsf{fma}\left(cosTheta, -1.5, -1\right), 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/ 1.0 (+ (+ c (/ t_0 cosTheta)) (fma t_0 (fma cosTheta -1.5 -1.0) 1.0)))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / ((c + (t_0 / cosTheta)) + fmaf(t_0, fmaf(cosTheta, -1.5f, -1.0f), 1.0f));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(c + Float32(t_0 / cosTheta)) + fma(t_0, fma(cosTheta, Float32(-1.5), Float32(-1.0)), Float32(1.0))))
end

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

6. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

   10. +-commutative N/A

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

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

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

8. Simplified 96.1%

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

Alternative 8: 96.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

   1. clear-num N/A

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

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

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

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

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

7. Applied egg-rr 96.0%

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

Alternative 9: 96.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

6. Taylor expanded in c around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

8. Simplified 58.4%

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

9. Taylor expanded in c around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. associate-+l+ N/A

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

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

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

    5. +-commutative N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. distribute-lft-out N/A

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

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

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

11. Simplified 95.7%

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

12. Final simplification 95.7%

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

Alternative 10: 96.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

6. Taylor expanded in c around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

8. Simplified 58.4%

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

9. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

    11. sub-neg N/A

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

    12. *-commutative N/A

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

    13. metadata-eval N/A

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

    14. accelerator-lowering-fma.f32 95.5

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

11. Simplified 95.5%

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

12. Final simplification 95.5%

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

Alternative 11: 95.7% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]
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 95.3%

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

Alternative 12: 94.9% 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 97.7%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + cosTheta \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}{cosTheta}}} \]
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 94.8

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

5. Simplified 94.8%

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

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. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. associate-+l+ N/A

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

   8. mul-1-neg N/A

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

   9. distribute-lft-neg-in N/A

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

   10. *-commutative N/A

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

   11. mul-1-neg N/A

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

   12. +-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. Simplified 94.6%

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

9. Final simplification 94.6%

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

Alternative 13: 92.8% accurate, 6.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (* cosTheta (fma (- 0.0 (* PI c)) cosTheta (sqrt PI))))

C

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

Julia

function code(cosTheta, c)
	return Float32(cosTheta * fma(Float32(Float32(0.0) - Float32(Float32(pi) * c)), cosTheta, sqrt(Float32(pi))))
end

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

5. Simplified 96.0%

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

6. Taylor expanded in c around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

8. Simplified 58.4%

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

9. Taylor expanded in cosTheta around 0

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

    1. mul-1-neg N/A

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

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

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

    3. associate-*l/ N/A

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

    4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

    9. *-commutative N/A

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

    10. *-lowering-*.f32 53.9

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

11. Simplified 53.9%

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

12. Taylor expanded in cosTheta around 0

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

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

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. mul-1-neg N/A

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

    7. distribute-lft-neg-in N/A

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

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

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

    9. *-commutative N/A

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

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

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

    11. mul-1-neg N/A

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

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

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

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

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

    14. mul-1-neg N/A

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

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

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

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

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

    17. PI-lowering-PI.f32 91.9

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

14. Simplified 91.9%

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

15. Final simplification 91.9%

    \[\leadsto cosTheta \cdot \mathsf{fma}\left(0 - \pi \cdot c, cosTheta, \sqrt{\pi}\right) \]
16. Add Preprocessing

Alternative 14: 92.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 91.9

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

5. Simplified 91.9%

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

Alternative 15: 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 97.7%

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

3. Taylor expanded in c around inf

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

5. Simplified 5.2%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))