Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 11.0s

Alternatives: 9

Speedup: 1.3×

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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((sqrt((single(1.0) - (cosTheta + cosTheta))) / (cosTheta * sqrt(single(pi)))) * exp((cosTheta * -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. 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+ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (((exp(cosTheta) ^ -cosTheta) * sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi)))) / 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. Taylor expanded in c around 0 97.7%

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

   1. associate-*l/ 98.1%

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

   2. mul-1-neg 98.1%

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

   3. unpow2 98.1%

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

   4. distribute-rgt-neg-out 98.1%

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

   5. exp-prod 98.1%

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

   6. cancel-sign-sub-inv 98.1%

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

   7. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 3: 98.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) * Float32(sqrt(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))) / 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 * single(-2.0))) / single(pi))) / 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. 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+ 98.7%

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

3. Applied egg-rr 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. associate-/r* 98.0%

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

   3. count-2 98.0%

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

   4. sqrt-div 98.1%

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

   5. cancel-sign-sub-inv 98.1%

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

   6. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

   1. *-lft-identity 98.1%

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

   2. metadata-eval 98.1%

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

   3. cancel-sign-sub-inv 98.1%

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

   4. cancel-sign-sub-inv 98.1%

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

   5. metadata-eval 98.1%

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

   6. *-commutative 98.1%

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

7. Simplified 98.1%

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

8. Final simplification 98.1%

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

Alternative 4: 96.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+ c (/ (- 1.0 cosTheta) (* (sqrt PI) (+ cosTheta (pow cosTheta 3.0))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((1.0f - cosTheta) / (sqrtf(((float) M_PI)) * (cosTheta + powf(cosTheta, 3.0f))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(1.0) - cosTheta) / Float32(sqrt(Float32(pi)) * Float32(cosTheta + (cosTheta ^ Float32(3.0))))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((single(1.0) - cosTheta) / (sqrt(single(pi)) * (cosTheta + (cosTheta ^ single(3.0)))))));
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. Step-by-step derivation

   1. associate-+l+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-commutative 97.7%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.7%

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

4. Taylor expanded in cosTheta around 0 96.5%

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

   1. neg-mul-1 96.5%

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

   2. sub-neg 96.5%

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

6. Simplified 96.5%

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

7. Taylor expanded in cosTheta around 0 96.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{1 - cosTheta}{\color{blue}{{cosTheta}^{3} \cdot \sqrt{\pi} + cosTheta \cdot \sqrt{\pi}}}\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-out 96.5%

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

9. Simplified 96.5%

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

10. Final simplification 96.5%

    \[\leadsto \frac{1}{1 + \left(c + \frac{1 - cosTheta}{\sqrt{\pi} \cdot \left(cosTheta + {cosTheta}^{3}\right)}\right)} \]

Alternative 5: 94.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (sqrt((single(1.0) / single(pi))) * (single(-1.0) + (single(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. Taylor expanded in c around 0 97.7%

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

   1. associate-*l/ 98.1%

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

   2. mul-1-neg 98.1%

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

   3. unpow2 98.1%

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

   4. distribute-rgt-neg-out 98.1%

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

   5. exp-prod 98.1%

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

   6. cancel-sign-sub-inv 98.1%

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

   7. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.0%

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

   2. exp-prod 98.0%

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

   3. distribute-rgt-neg-in 98.0%

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

   4. unpow2 98.0%

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

   5. rec-exp 98.1%

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

   6. unpow2 98.1%

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

   7. exp-prod 98.1%

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

   8. times-frac 98.1%

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

   9. *-lft-identity 98.1%

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

   10. +-commutative 98.1%

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

   11. *-commutative 98.1%

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

   12. fma-def 98.1%

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

   13. *-commutative 98.1%

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

6. Simplified 98.1%

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

   1. add-log-exp 98.1%

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

8. Applied egg-rr 98.1%

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

9. Taylor expanded in cosTheta around 0 95.2%

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

    1. distribute-rgt-out 95.2%

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

11. Simplified 95.2%

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

12. Final simplification 95.2%

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

Alternative 6: 92.9% accurate, 2.1× 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. 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+ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in cosTheta around 0 94.0%

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

5. Final simplification 94.0%

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

Alternative 7: 10.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = c * (single(pi) * (cosTheta * -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. Step-by-step derivation

   1. associate-+l+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-commutative 97.7%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.7%

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

4. Taylor expanded in cosTheta around 0 93.6%

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

5. Taylor expanded in cosTheta around 0 93.7%

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

   1. mul-1-neg 93.7%

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

   2. unsub-neg 93.7%

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

   3. *-commutative 93.7%

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

   4. *-commutative 93.7%

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

   5. associate-*l* 93.7%

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

   6. unpow2 93.7%

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

   7. +-commutative 93.7%

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

7. Simplified 93.7%

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

8. Taylor expanded in c around inf 10.9%

   \[\leadsto \color{blue}{-1 \cdot \left(c \cdot \left({cosTheta}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 10.9%

      \[\leadsto \color{blue}{-c \cdot \left({cosTheta}^{2} \cdot \pi\right)} \]

   2. distribute-rgt-neg-in 10.9%

      \[\leadsto \color{blue}{c \cdot \left(-{cosTheta}^{2} \cdot \pi\right)} \]

   3. *-commutative 10.9%

      \[\leadsto c \cdot \left(-\color{blue}{\pi \cdot {cosTheta}^{2}}\right) \]

   4. distribute-rgt-neg-in 10.9%

      \[\leadsto c \cdot \color{blue}{\left(\pi \cdot \left(-{cosTheta}^{2}\right)\right)} \]

   5. unpow2 10.9%

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

   6. distribute-rgt-neg-in 10.9%

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

10. Simplified 10.9%

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

11. Final simplification 10.9%

    \[\leadsto c \cdot \left(\pi \cdot \left(cosTheta \cdot \left(-cosTheta\right)\right)\right) \]

Alternative 8: 10.8% accurate, 140.3× speedup

Math

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

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. Step-by-step derivation

   1. associate-+l+ 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-commutative 97.7%

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

   5. associate-/r/ 98.0%

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

3. Simplified 98.7%

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

4. Taylor expanded in cosTheta around inf 10.7%

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

5. Taylor expanded in c around 0 10.7%

   \[\leadsto \color{blue}{-1 \cdot c + 1} \]
6. Step-by-step derivation

   1. +-commutative 10.7%

      \[\leadsto \color{blue}{1 + -1 \cdot c} \]

   2. mul-1-neg 10.7%

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

   3. unsub-neg 10.7%

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

7. Simplified 10.7%

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

8. Final simplification 10.7%

   \[\leadsto 1 - c \]

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 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. Taylor expanded in c around 0 97.7%

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

3. Taylor expanded in cosTheta around inf 10.7%

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

4. Final simplification 10.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023228 
(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))))))