Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.4%

Time: 10.3s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

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(1.0) / Float32(sqrt(Float32(pi)) / Float32(sqrt(Float32(Float32(1.0) - Float32(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

Derivation

1. Initial program 98.2%

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

   1. frac-times 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. associate--r+ 98.6%

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

   4. associate-/l/ 98.5%

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

   5. clear-num 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. frac-times 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. associate--r+ 98.6%

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

   4. *-commutative 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-cosTheta \cdot cosTheta}}{cosTheta}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))) * (exp(-(cosTheta * cosTheta)) / cosTheta)));
end

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in c around 0 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} + 1}} \]
3. Step-by-step derivation

   1. associate-*l/ 98.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{-1 \cdot {cosTheta}^{2}} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}{cosTheta}} + 1} \]

   2. mul-1-neg 98.0%

      \[\leadsto \frac{1}{\frac{e^{\color{blue}{-{cosTheta}^{2}}} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}{cosTheta} + 1} \]

   3. unpow2 98.0%

      \[\leadsto \frac{1}{\frac{e^{-\color{blue}{cosTheta \cdot cosTheta}} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}{cosTheta} + 1} \]

   4. distribute-lft-neg-out 98.0%

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

   5. *-commutative 98.0%

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

   6. cancel-sign-sub-inv 98.0%

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

   7. metadata-eval 98.0%

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

4. Applied egg-rr 98.0%

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

   1. distribute-rgt-neg-out 98.0%

      \[\leadsto \frac{1}{\frac{e^{\color{blue}{-cosTheta \cdot cosTheta}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}}{cosTheta} + 1} \]

   2. unpow2 98.0%

      \[\leadsto \frac{1}{\frac{e^{-\color{blue}{{cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}}{cosTheta} + 1} \]

   3. mul-1-neg 98.0%

      \[\leadsto \frac{1}{\frac{e^{\color{blue}{-1 \cdot {cosTheta}^{2}}} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}}{cosTheta} + 1} \]

   4. metadata-eval 98.0%

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

   5. cancel-sign-sub-inv 98.0%

      \[\leadsto \frac{1}{\frac{e^{-1 \cdot {cosTheta}^{2}} \cdot \sqrt{\frac{\color{blue}{1 - 2 \cdot cosTheta}}{\pi}}}{cosTheta} + 1} \]

   6. associate-*l/ 98.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} + 1} \]

   7. *-commutative 98.2%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta}} + 1} \]

   8. cancel-sign-sub-inv 98.2%

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

   9. metadata-eval 98.2%

      \[\leadsto \frac{1}{\sqrt{\frac{1 + \color{blue}{-2} \cdot cosTheta}{\pi}} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} + 1} \]

   10. *-commutative 98.2%

       \[\leadsto \frac{1}{\sqrt{\frac{1 + \color{blue}{cosTheta \cdot -2}}{\pi}} \cdot \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} + 1} \]

   11. mul-1-neg 98.2%

       \[\leadsto \frac{1}{\sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{\color{blue}{-{cosTheta}^{2}}}}{cosTheta} + 1} \]

   12. unpow2 98.2%

       \[\leadsto \frac{1}{\sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-\color{blue}{cosTheta \cdot cosTheta}}}{cosTheta} + 1} \]

6. Simplified 98.2%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-cosTheta \cdot cosTheta}}{cosTheta}} + 1} \]

7. Final simplification 98.2%

   \[\leadsto \frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{-cosTheta \cdot cosTheta}}{cosTheta}} \]

Alternative 4: 97.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. associate-+l+ 98.2%

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

   2. associate-*l* 98.2%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 96.7%

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

   1. +-commutative 96.7%

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

   2. associate--l+ 96.7%

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

   3. *-commutative 96.7%

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

   4. fma-neg 96.7%

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

   5. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Final simplification 96.7%

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

Alternative 5: 96.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(-1.0) + ((single(1.0) / cosTheta) + (cosTheta * single(-1.5)))) * sqrt((single(1.0) / single(pi)))));
end

Derivation

1. Initial program 98.2%

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

   1. associate-+l+ 98.2%

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

   2. associate-*l* 98.2%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 96.7%

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

   1. +-commutative 96.7%

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

   2. associate--l+ 96.7%

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

   3. *-commutative 96.7%

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

   4. fma-neg 96.7%

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

   5. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in c around 0 96.2%

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

8. Final simplification 96.2%

   \[\leadsto \frac{1}{1 + \left(-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]

Alternative 6: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (((cosTheta * single(-1.5)) + ((single(1.0) / cosTheta) + single(-1.0))) / sqrt(single(pi)))));
end

Derivation

1. Initial program 98.2%

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

   1. associate-+l+ 98.2%

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

   2. associate-*l* 98.2%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 96.7%

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

   1. associate--l+ 96.7%

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

   2. *-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

7. Final simplification 96.7%

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

Alternative 7: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. associate-+l+ 98.2%

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

   2. associate-*l* 98.2%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 95.0%

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

5. Final simplification 95.0%

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

Alternative 8: 93.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.0%

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

   2. distribute-lft-neg-out 98.0%

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

   3. exp-neg 98.1%

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

   4. times-frac 98.0%

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

   5. *-commutative 98.0%

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

   6. times-frac 98.1%

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

   7. associate-*l/ 98.1%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 92.0%

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

5. Final simplification 92.0%

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

Alternative 9: 10.8% accurate, 421.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in c around 0 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}} + 1}} \]

3. Taylor expanded in cosTheta around inf 11.2%

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

4. Final simplification 11.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(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))))))