Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 19.3s

Alternatives: 9

Speedup: 0.6×

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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. fma-define 97.5%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 98.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. fma-define 97.5%

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

3. Simplified 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. associate-/r* 97.8%

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

   3. sqrt-undiv 97.9%

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

   4. +-commutative 97.9%

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

   5. fma-define 97.9%

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

6. Applied egg-rr 97.9%

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

   1. *-lft-identity 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.5%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate--l- 98.1%

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

   6. sub-neg 98.1%

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

   7. neg-mul-1 98.1%

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

   8. distribute-lft-out 98.1%

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

   9. distribute-rgt-out 98.1%

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

   10. metadata-eval 98.1%

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

   11. distribute-lft-neg-out 98.1%

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

   12. distribute-rgt-neg-out 98.1%

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

   13. exp-prod 98.1%

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

3. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

   3. +-commutative 98.1%

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

   4. fma-define 98.1%

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

6. Applied egg-rr 98.1%

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

   1. unpow-1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in c around 0 97.4%

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

    1. associate-*l/ 97.7%

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

    2. neg-mul-1 97.7%

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

    3. +-commutative 97.7%

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

    4. *-commutative 97.7%

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

    5. fma-undefine 97.7%

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

11. Simplified 97.7%

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

12. Taylor expanded in cosTheta around inf 97.7%

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

    1. associate-*r/ 97.7%

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

    2. metadata-eval 97.7%

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

14. Simplified 97.7%

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

15. Final simplification 97.7%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.5%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate--l- 98.1%

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

   6. sub-neg 98.1%

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

   7. neg-mul-1 98.1%

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

   8. distribute-lft-out 98.1%

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

   9. distribute-rgt-out 98.1%

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

   10. metadata-eval 98.1%

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

   11. distribute-lft-neg-out 98.1%

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

   12. distribute-rgt-neg-out 98.1%

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

   13. exp-prod 98.1%

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

3. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

   3. +-commutative 98.1%

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

   4. fma-define 98.1%

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

6. Applied egg-rr 98.1%

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

   1. unpow-1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in c around 0 97.4%

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

    1. associate-*l/ 97.7%

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

    2. neg-mul-1 97.7%

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

    3. +-commutative 97.7%

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

    4. *-commutative 97.7%

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

    5. fma-undefine 97.7%

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

11. Simplified 97.7%

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

12. Taylor expanded in cosTheta around 0 97.7%

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

13. Final simplification 97.7%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.5%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate--l- 98.1%

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

   6. sub-neg 98.1%

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

   7. neg-mul-1 98.1%

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

   8. distribute-lft-out 98.1%

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

   9. distribute-rgt-out 98.1%

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

   10. metadata-eval 98.1%

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

   11. distribute-lft-neg-out 98.1%

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

   12. distribute-rgt-neg-out 98.1%

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

   13. exp-prod 98.1%

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

3. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

   3. +-commutative 98.1%

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

   4. fma-define 98.1%

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

6. Applied egg-rr 98.1%

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

   1. unpow-1 98.1%

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

8. Simplified 98.1%

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

   1. inv-pow 98.1%

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

   2. add-sqr-sqrt 97.3%

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

   3. unpow-prod-down 97.2%

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

10. Applied egg-rr 97.2%

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

    1. pow-sqr 97.7%

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

    2. associate-*l/ 97.7%

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

    3. *-lft-identity 97.7%

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

    4. metadata-eval 97.7%

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

12. Simplified 97.7%

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

13. Taylor expanded in c around 0 97.4%

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

14. Final simplification 97.4%

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

Alternative 6: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. fma-define 97.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. *-commutative 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. mul-1-neg 96.7%

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

   7. unsub-neg 96.7%

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

   8. *-commutative 96.7%

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

7. Simplified 96.7%

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

8. Final simplification 96.7%

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

Alternative 7: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.5%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate--l- 98.1%

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

   6. sub-neg 98.1%

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

   7. neg-mul-1 98.1%

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

   8. distribute-lft-out 98.1%

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

   9. distribute-rgt-out 98.1%

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

   10. metadata-eval 98.1%

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

   11. distribute-lft-neg-out 98.1%

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

   12. distribute-rgt-neg-out 98.1%

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

   13. exp-prod 98.1%

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

3. Simplified 98.1%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

   3. +-commutative 98.1%

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

   4. fma-define 98.1%

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

6. Applied egg-rr 98.1%

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

   1. unpow-1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in c around 0 97.4%

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

    1. associate-*l/ 97.7%

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

    2. neg-mul-1 97.7%

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

    3. +-commutative 97.7%

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

    4. *-commutative 97.7%

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

    5. fma-undefine 97.7%

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

11. Simplified 97.7%

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

12. Taylor expanded in cosTheta around 0 96.5%

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

    1. mul-1-neg 96.5%

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

    2. unsub-neg 96.5%

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

    3. associate-*r* 96.5%

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

    4. mul-1-neg 96.5%

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

    5. unsub-neg 96.5%

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

14. Simplified 96.5%

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

15. Final simplification 96.5%

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

Alternative 8: 93.2% accurate, 3.1× 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. +-commutative 97.5%

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

   3. fma-define 97.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 94.4%

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

6. Final simplification 94.4%

   \[\leadsto cosTheta \cdot \sqrt{\pi} \]
7. Add Preprocessing

Alternative 9: 10.7% accurate, 107.3× 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 97.5%

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

   1. *-un-lft-identity 97.5%

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

   2. add-cube-cbrt 97.5%

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

   3. prod-diff 97.5%

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

   4. sub-neg 97.5%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 93.1%

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

   7. sqr-neg 93.1%

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

   8. sqrt-unprod 93.1%

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

   9. add-sqr-sqrt 93.1%

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

   10. pow2 93.1%

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

   11. pow2 93.1%

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

   12. pow2 93.1%

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

4. Applied egg-rr 93.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(1, 1 + cosTheta, -\sqrt[3]{cosTheta} \cdot {\left(\sqrt[3]{cosTheta}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{cosTheta}, {\left(\sqrt[3]{cosTheta}\right)}^{2}, \sqrt[3]{cosTheta} \cdot {\left(\sqrt[3]{cosTheta}\right)}^{2}\right)}}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. fma-undefine 93.1%

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

   2. *-lft-identity 93.1%

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

   3. associate-+l+ 93.1%

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

   4. metadata-eval 93.1%

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

   5. rem-square-sqrt 93.1%

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

   6. rem-square-sqrt 93.1%

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

   7. hypot-undefine 93.1%

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

   8. hypot-undefine 93.1%

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

   9. associate-+l+ 93.1%

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

6. Simplified 93.1%

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

7. Taylor expanded in cosTheta around inf 10.2%

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

8. Taylor expanded in c around 0 10.2%

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

   1. mul-1-neg 10.2%

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

   2. unsub-neg 10.2%

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

10. Simplified 10.2%

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

11. Final simplification 10.2%

    \[\leadsto 1 - c \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))))))