Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 18.1s

Alternatives: 12

Speedup: 1.5×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. associate-+r+ 98.0%

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

   3. +-commutative 98.0%

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

   4. fma-def 98.0%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 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 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-def 98.0%

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

   4. times-frac 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate--l- 98.6%

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

   7. count-2 98.6%

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

   8. *-commutative 98.6%

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

   9. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. div-inv 98.3%

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

   4. associate-/l* 98.4%

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

   5. associate--l- 98.4%

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

   6. count-2 98.4%

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

   7. cancel-sign-sub-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. *-commutative 98.4%

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

   10. +-commutative 98.4%

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

   11. fma-udef 98.4%

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

3. Applied egg-rr 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. add-sqr-sqrt 97.7%

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

   3. times-frac 97.5%

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

   4. frac-2neg 97.5%

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

   5. frac-2neg 97.5%

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

   6. associate-/r/ 97.8%

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

   7. /-rgt-identity 97.8%

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

   8. frac-2neg 97.8%

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

   9. frac-2neg 97.8%

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

   10. associate-/r/ 97.6%

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

   11. *-commutative 97.6%

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

   12. /-rgt-identity 97.6%

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

5. Applied egg-rr 97.6%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.6%

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

   3. associate-/l* 97.7%

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

   4. rem-square-sqrt 98.5%

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

   5. associate-/r/ 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-/r* 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 4: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. div-inv 98.3%

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

   4. associate-/l* 98.4%

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

   5. associate--l- 98.4%

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

   6. count-2 98.4%

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

   7. cancel-sign-sub-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. *-commutative 98.4%

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

   10. +-commutative 98.4%

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

   11. fma-udef 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. frac-times 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. associate--l- 98.6%

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

   4. count-2 98.6%

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

   5. cancel-sign-sub-inv 98.6%

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

   6. metadata-eval 98.6%

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

   7. *-commutative 98.6%

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

   8. +-commutative 98.6%

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

   9. fma-udef 98.6%

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

   10. associate-/l/ 98.3%

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

3. Applied egg-rr 98.3%

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

4. Final simplification 98.3%

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

Alternative 6: 98.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. div-inv 98.3%

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

   4. associate-/l* 98.4%

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

   5. associate--l- 98.4%

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

   6. count-2 98.4%

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

   7. cancel-sign-sub-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. *-commutative 98.4%

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

   10. +-commutative 98.4%

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

   11. fma-udef 98.4%

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

3. Applied egg-rr 98.4%

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

   1. associate-/r/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. un-div-inv 98.2%

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

5. Applied egg-rr 98.3%

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

   1. *-lft-identity 98.3%

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

7. Simplified 98.3%

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

   1. frac-2neg 98.3%

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

   2. distribute-frac-neg 98.3%

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

9. Applied egg-rr 98.3%

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

    1. distribute-neg-frac 98.3%

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

    2. neg-sub0 98.3%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\frac{\color{blue}{0 - \mathsf{fma}\left(cosTheta, -2, 1\right)}}{-\pi}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    3. fma-udef 98.3%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\frac{0 - \color{blue}{\left(cosTheta \cdot -2 + 1\right)}}{-\pi}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    4. *-commutative 98.3%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\frac{0 - \left(\color{blue}{-2 \cdot cosTheta} + 1\right)}{-\pi}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    5. +-commutative 98.3%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\frac{0 - \color{blue}{\left(1 + -2 \cdot cosTheta\right)}}{-\pi}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    6. associate--r+ 98.3%

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\frac{\color{blue}{\left(0 - 1\right) - -2 \cdot cosTheta}}{-\pi}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

    7. metadata-eval 98.3%

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

    8. *-commutative 98.3%

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 7: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. div-inv 98.3%

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

   4. associate-/l* 98.4%

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

   5. associate--l- 98.4%

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

   6. count-2 98.4%

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

   7. cancel-sign-sub-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. *-commutative 98.4%

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

   10. +-commutative 98.4%

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

   11. fma-udef 98.4%

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

3. Applied egg-rr 98.4%

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

4. Taylor expanded in cosTheta around 0 95.2%

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

   1. distribute-rgt-out 95.1%

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

6. Simplified 95.1%

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

   1. *-commutative 95.1%

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

   2. sqrt-div 95.1%

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

   3. metadata-eval 95.1%

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

   4. un-div-inv 95.5%

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

8. Applied egg-rr 95.5%

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

9. Final simplification 95.5%

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

Alternative 8: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. frac-times 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. associate--l- 98.6%

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

   4. count-2 98.6%

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

   5. cancel-sign-sub-inv 98.6%

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

   6. metadata-eval 98.6%

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

   7. *-commutative 98.6%

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

   8. +-commutative 98.6%

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

   9. fma-udef 98.6%

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

   10. associate-/l/ 98.3%

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

3. Applied egg-rr 98.3%

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

4. Taylor expanded in cosTheta around 0 95.5%

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

   1. mul-1-neg 95.5%

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

6. Simplified 95.5%

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

7. Final simplification 95.5%

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

Alternative 9: 95.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.0%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. sub-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. associate-+l+ 98.3%

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

   8. count-2 98.3%

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

   9. *-commutative 98.3%

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

   10. distribute-lft-neg-in 98.3%

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

   11. distribute-rgt-neg-in 98.3%

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

   12. metadata-eval 98.3%

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

   13. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around 0 94.7%

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

   1. sqrt-div 94.7%

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

   2. metadata-eval 94.7%

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

   3. frac-times 95.3%

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

   4. metadata-eval 95.3%

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

6. Applied egg-rr 95.3%

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

7. Final simplification 95.3%

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

Alternative 10: 94.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.0%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. sub-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. associate-+l+ 98.3%

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

   8. count-2 98.3%

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

   9. *-commutative 98.3%

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

   10. distribute-lft-neg-in 98.3%

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

   11. distribute-rgt-neg-in 98.3%

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

   12. metadata-eval 98.3%

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

   13. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around 0 94.7%

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

5. Taylor expanded in c around 0 94.8%

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

   1. distribute-rgt-out 94.7%

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

7. Simplified 94.7%

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

8. Final simplification 94.7%

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

Alternative 11: 95.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.0%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. sub-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. associate-+l+ 98.3%

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

   8. count-2 98.3%

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

   9. *-commutative 98.3%

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

   10. distribute-lft-neg-in 98.3%

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

   11. distribute-rgt-neg-in 98.3%

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

   12. metadata-eval 98.3%

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

   13. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around 0 94.7%

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

   1. div-inv 94.7%

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

   2. distribute-rgt-out 94.7%

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

   3. sqrt-div 94.7%

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

   4. metadata-eval 94.7%

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

   5. associate-*l/ 95.0%

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

   6. *-un-lft-identity 95.0%

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

   7. +-commutative 95.0%

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

6. Applied egg-rr 95.0%

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

   1. *-lft-identity 95.0%

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

   2. associate-+r+ 95.0%

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

   3. +-commutative 95.0%

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

   4. associate-+l+ 95.0%

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

   5. +-commutative 95.0%

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 12: 10.8% accurate, 322.0× 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 98.0%

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

   1. +-commutative 98.0%

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

   2. associate-+r+ 98.0%

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

   3. +-commutative 98.0%

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

   4. fma-def 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 10.7%

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

5. Taylor expanded in c around 0 10.7%

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

6. Final simplification 10.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(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))))))