Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.0%

Time: 11.8s

Alternatives: 10

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: 98.0% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 2: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 98.2%

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

Alternative 3: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 98.0%

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

10. Applied rewrites 98.0%

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

11. Final simplification 98.0%

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

Alternative 4: 97.0% 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 98.1%

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

3. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 97.7%

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

Alternative 5: 96.8% accurate, 2.9× speedup

Math

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

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

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);
}

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(1.0)))
end

Derivation

1. Initial program 98.1%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f32 N/A

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 97.0%

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

Alternative 6: 95.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower--.f32 95.8

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

5. Applied rewrites 95.8%

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

7. Applied rewrites 96.5%

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

Alternative 7: 95.9% 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 98.1%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 96.4%

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

Alternative 8: 95.8% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in c around inf

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

8. Applied rewrites 93.6%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 96.1%

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

Alternative 9: 93.1% 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 98.1%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-PI.f32 93.7

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

5. Applied rewrites 93.7%

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

Alternative 10: 5.1% 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 98.1%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f32 5.0

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

5. Applied rewrites 5.0%

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

Reproduce

herbie shell --seed 2024222 
(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))))))