Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.4%

Time: 12.5s

Alternatives: 11

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: 97.9% 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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / cosTheta) * (exp(-cosTheta) ^ cosTheta)) / sqrt((single(pi) / (single(1.0) + (cosTheta * single(-2.0)))))));
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. Step-by-step derivation

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. associate--r+ 98.5%

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

   4. associate-/l/ 98.6%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 97.7%

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

   2. associate-/r/ 97.7%

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

   3. associate-/r/ 97.6%

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

5. Applied egg-rr 97.6%

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

   1. *-un-lft-identity 97.6%

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

   2. add-sqr-sqrt 98.4%

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

   3. *-commutative 98.4%

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

   4. sqrt-undiv 98.6%

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

7. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

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

   2. associate-/r* 98.6%

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

   3. count-2 98.6%

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

   4. cancel-sign-sub-inv 98.6%

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

   5. metadata-eval 98.6%

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

9. Simplified 98.6%

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

    1. *-un-lft-identity 98.6%

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

    2. associate-+l+ 98.6%

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

    3. associate-*l/ 98.7%

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

    4. exp-prod 98.7%

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

    5. *-commutative 98.7%

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

11. Applied egg-rr 98.7%

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

    1. *-lft-identity 98.7%

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

    2. associate-+r+ 98.7%

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

13. Simplified 98.7%

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

14. Final simplification 98.7%

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

Alternative 2: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{\left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + 0.5 \cdot {cosTheta}^{2}\right)\right) + -1}{\sqrt{\pi}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (+
      (+ (* cosTheta -1.5) (+ (/ 1.0 cosTheta) (* 0.5 (pow cosTheta 2.0))))
      -1.0)
     (sqrt PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((((cosTheta * -1.5f) + ((1.0f / cosTheta) + (0.5f * powf(cosTheta, 2.0f)))) + -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(cosTheta * Float32(-1.5)) + Float32(Float32(Float32(1.0) / cosTheta) + Float32(Float32(0.5) * (cosTheta ^ Float32(2.0))))) + 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(0.5) * (cosTheta ^ single(2.0))))) + single(-1.0)) / 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. Step-by-step derivation

   1. associate-+l+ 98.1%

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

   2. associate-*l* 98.1%

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

   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 97.6%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1}}{\sqrt{\pi}}\right)} \]

5. Final simplification 97.6%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + 0.5 \cdot {cosTheta}^{2}\right)\right) + -1}{\sqrt{\pi}}\right)} \]

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp((cosTheta * -cosTheta)) * (single(1.0) / (cosTheta * sqrt((single(pi) / (single(1.0) + (cosTheta * single(-2.0)))))))));
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. Step-by-step derivation

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. associate--r+ 98.5%

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

   4. associate-/l/ 98.6%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 97.7%

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

   2. associate-/r/ 97.7%

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

   3. associate-/r/ 97.6%

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

5. Applied egg-rr 97.6%

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

   1. pow2 97.6%

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

   2. *-commutative 97.6%

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

   3. sqrt-undiv 97.6%

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

7. Applied egg-rr 97.6%

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

   1. unpow2 97.6%

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

   2. rem-square-sqrt 98.6%

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

   3. sub-neg 98.6%

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

   4. count-2 98.6%

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

   5. *-commutative 98.6%

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

   6. distribute-rgt-neg-in 98.6%

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

   7. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / cosTheta) / sqrtf((((float) M_PI) / (1.0f + (cosTheta * -2.0f))))) * 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) / cosTheta) / sqrt(Float32(Float32(pi) / Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0)))))) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / cosTheta) / sqrt((single(pi) / (single(1.0) + (cosTheta * single(-2.0)))))) * exp((cosTheta * -cosTheta))));
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. Step-by-step derivation

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. associate--r+ 98.5%

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

   4. associate-/l/ 98.6%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 97.7%

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

   2. associate-/r/ 97.7%

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

   3. associate-/r/ 97.6%

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

5. Applied egg-rr 97.6%

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

   1. *-un-lft-identity 97.6%

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

   2. add-sqr-sqrt 98.4%

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

   3. *-commutative 98.4%

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

   4. sqrt-undiv 98.6%

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

7. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

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

   2. associate-/r* 98.6%

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

   3. count-2 98.6%

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

   4. cancel-sign-sub-inv 98.6%

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

   5. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 5: 97.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (/
    (/ (sqrt (- 1.0 (+ cosTheta cosTheta))) cosTheta)
    (* (sqrt PI) (+ 1.0 (* 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)) * (1.0f + (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))) / cosTheta) / Float32(sqrt(Float32(pi)) * Float32(Float32(1.0) + Float32(cosTheta * 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)) * (single(1.0) + (cosTheta * cosTheta)))));
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. Step-by-step derivation

   1. +-commutative 98.1%

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

   2. fma-def 98.1%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate--l- 98.6%

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

   6. *-commutative 98.6%

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

   7. exp-prod 98.6%

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

3. Simplified 98.6%

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

   1. fma-udef 98.6%

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

   2. pow-neg 98.6%

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

   3. div-inv 98.6%

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

   4. associate-/l/ 98.5%

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

   5. pow-exp 98.5%

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

5. Applied egg-rr 98.5%

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

6. Taylor expanded in cosTheta around 0 97.5%

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

   1. unpow2 97.5%

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

   2. distribute-rgt1-in 97.5%

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 6: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((((1.0f / cosTheta) + (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(Float32(1.0) / cosTheta) + Float32(cosTheta * Float32(-1.5))) + 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) + (cosTheta * single(-1.5))) + single(-1.0)) / 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. Step-by-step derivation

   1. associate-+l+ 98.1%

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

   2. associate-*l* 98.1%

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

   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 96.9%

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

5. Final simplification 96.9%

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

Alternative 7: 94.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. +-commutative 98.1%

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

   2. fma-def 98.1%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate--l- 98.6%

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

   6. *-commutative 98.6%

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

   7. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 94.7%

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

   1. associate-+r+ 94.7%

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

   2. +-commutative 94.7%

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

   3. distribute-rgt-out 94.7%

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

6. Simplified 94.7%

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

7. Final simplification 94.7%

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

Alternative 8: 94.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. associate-*l* 98.1%

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

   2. *-lft-identity 98.1%

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

   3. associate-*r* 98.1%

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

   4. *-lft-identity 98.1%

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

   5. associate--l- 98.0%

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

   6. *-commutative 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in cosTheta around 0 96.4%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(-0.5 \cdot cosTheta + \frac{1}{cosTheta}\right) - 1\right)} \cdot e^{cosTheta \cdot \left(-cosTheta\right)}\right)} \]

5. Taylor expanded in cosTheta around 0 96.4%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left({cosTheta}^{2} + \left(-1.5 \cdot cosTheta + \frac{1}{cosTheta}\right)\right) - 1\right)}} \]
6. Step-by-step derivation

   1. associate--l+ 96.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left({cosTheta}^{2} + \left(\left(-1.5 \cdot cosTheta + \frac{1}{cosTheta}\right) - 1\right)\right)}} \]

   2. unpow2 96.4%

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

   3. sub-neg 96.4%

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

   4. fma-def 96.4%

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

   5. metadata-eval 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in c around 0 96.3%

   \[\leadsto \color{blue}{\frac{1}{1 + \left(\left({cosTheta}^{2} + \left(-1.5 \cdot cosTheta + \frac{1}{cosTheta}\right)\right) - 1\right) \cdot \sqrt{\frac{1}{\pi}}}} \]

9. Taylor expanded in cosTheta around 0 94.6%

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

10. Final simplification 94.6%

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

Alternative 9: 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 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. Step-by-step derivation

   1. +-commutative 98.1%

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

   2. fma-def 98.1%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate--l- 98.6%

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

   6. *-commutative 98.6%

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

   7. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 92.0%

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

5. Final simplification 92.0%

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

Alternative 10: 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 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. Step-by-step derivation

   1. +-commutative 98.1%

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

   2. fma-def 98.1%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate--l- 98.6%

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

   6. *-commutative 98.6%

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

   7. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 10.9%

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

5. Taylor expanded in c around 0 10.9%

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

   1. +-commutative 10.9%

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

   2. mul-1-neg 10.9%

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

   3. unsub-neg 10.9%

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

7. Simplified 10.9%

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

8. Final simplification 10.9%

   \[\leadsto 1 - c \]

Alternative 11: 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 98.1%

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

   1. +-commutative 98.1%

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

   2. fma-def 98.1%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. associate--l- 98.6%

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

   6. *-commutative 98.6%

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

   7. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 10.9%

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

5. Taylor expanded in c around 0 10.9%

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

6. Final simplification 10.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023242 
(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))))))