Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.3%

Time: 18.1s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (/ 1.0 (/ (sqrt PI) (/ (sqrt (- 1.0 (* 2.0 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 - (2.0f * cosTheta))) / cosTheta))) * expf((cosTheta * -cosTheta))));
}

Julia

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

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.5%

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

   2. pow2 97.5%

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

   3. frac-times 98.0%

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

   4. *-un-lft-identity 98.0%

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

   5. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. add-sqr-sqrt 98.5%

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

   3. associate-/l/ 98.5%

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

   4. clear-num 98.7%

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

   5. count-2 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. frac-times 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. associate--l- 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 97.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * (1.0f / (sqrtf(((float) M_PI)) / ((1.0f + (cosTheta * ((cosTheta * ((cosTheta * -0.5f) - 0.5f)) + -1.0f))) / cosTheta)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) * Float32(Float32(1.0) / Float32(sqrt(Float32(pi)) / Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(Float32(cosTheta * Float32(-0.5)) - Float32(0.5))) + Float32(-1.0)))) / cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp((cosTheta * -cosTheta)) * (single(1.0) / (sqrt(single(pi)) / ((single(1.0) + (cosTheta * ((cosTheta * ((cosTheta * single(-0.5)) - single(0.5))) + single(-1.0)))) / cosTheta)))));
end

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.5%

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

   2. pow2 97.5%

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

   3. frac-times 98.0%

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

   4. *-un-lft-identity 98.0%

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

   5. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. add-sqr-sqrt 98.5%

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

   3. associate-/l/ 98.5%

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

   4. clear-num 98.7%

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

   5. count-2 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in cosTheta around 0 97.7%

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

8. Final simplification 97.7%

   \[\leadsto \frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{1}{\frac{\sqrt{\pi}}{\frac{1 + cosTheta \cdot \left(cosTheta \cdot \left(cosTheta \cdot -0.5 - 0.5\right) + -1\right)}{cosTheta}}}} \]
9. Add Preprocessing

Alternative 4: 97.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * (((1.0f + (cosTheta * ((cosTheta * ((cosTheta * -0.5f) - 0.5f)) + -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(Float32(Float32(1.0) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(Float32(cosTheta * Float32(-0.5)) - Float32(0.5))) + Float32(-1.0)))) / 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 * ((cosTheta * single(-0.5)) - single(0.5))) + single(-1.0)))) / cosTheta) / sqrt(single(pi)))));
end

Derivation

1. Initial program 97.9%

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

   1. 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. associate--l- 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in cosTheta around 0 97.5%

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

6. Final simplification 97.5%

   \[\leadsto \frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{\frac{1 + cosTheta \cdot \left(cosTheta \cdot \left(cosTheta \cdot -0.5 - 0.5\right) + -1\right)}{cosTheta}}{\sqrt{\pi}}} \]
7. Add Preprocessing

Alternative 5: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (expf((cosTheta * -cosTheta)) * (1.0f / (sqrtf(((float) M_PI)) / ((1.0f + (cosTheta * ((cosTheta * -0.5f) + -1.0f))) / cosTheta)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) * Float32(Float32(1.0) / Float32(sqrt(Float32(pi)) / Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(-0.5)) + Float32(-1.0)))) / cosTheta))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp((cosTheta * -cosTheta)) * (single(1.0) / (sqrt(single(pi)) / ((single(1.0) + (cosTheta * ((cosTheta * single(-0.5)) + single(-1.0)))) / cosTheta)))));
end

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.5%

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

   2. pow2 97.5%

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

   3. frac-times 98.0%

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

   4. *-un-lft-identity 98.0%

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

   5. associate--l- 98.0%

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

4. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

   2. add-sqr-sqrt 98.5%

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

   3. associate-/l/ 98.5%

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

   4. clear-num 98.7%

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

   5. count-2 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in cosTheta around 0 97.3%

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

8. Final simplification 97.3%

   \[\leadsto \frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{1}{\frac{\sqrt{\pi}}{\frac{1 + cosTheta \cdot \left(cosTheta \cdot -0.5 + -1\right)}{cosTheta}}}} \]
9. Add Preprocessing

Alternative 6: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) - (expf((cosTheta * -cosTheta)) * (((-1.0f - (cosTheta * ((cosTheta * -0.5f) + -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(Float32(Float32(-1.0) - Float32(cosTheta * Float32(Float32(cosTheta * Float32(-0.5)) + Float32(-1.0)))) / 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 * single(-0.5)) + single(-1.0)))) / cosTheta) / sqrt(single(pi)))));
end

Derivation

1. Initial program 97.9%

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

   1. 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. associate--l- 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in cosTheta around 0 97.1%

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

6. Final simplification 97.1%

   \[\leadsto \frac{1}{\left(1 + c\right) - e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{\frac{-1 - cosTheta \cdot \left(cosTheta \cdot -0.5 + -1\right)}{cosTheta}}{\sqrt{\pi}}} \]
7. Add Preprocessing

Alternative 7: 95.8% accurate, 1.5× speedup

Math

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

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 97.9%

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

   1. 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. associate--l- 98.5%

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

4. Applied egg-rr 98.5%

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

5. Taylor expanded in cosTheta around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 8: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ 97.9%

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

   2. distribute-lft-neg-out 97.9%

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

   3. distribute-rgt-neg-out 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in cosTheta around 0 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. associate-*r* 95.9%

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

   4. mul-1-neg 95.9%

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

   5. unsub-neg 95.9%

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

7. Simplified 95.9%

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

8. Final simplification 95.9%

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

Alternative 9: 95.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (- (+ 1.0 (+ c (/ (pow PI -0.5) cosTheta))) (pow PI -0.5))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + (c + (powf(((float) M_PI), -0.5f) / cosTheta))) - powf(((float) M_PI), -0.5f));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(c + Float32((Float32(pi) ^ Float32(-0.5)) / cosTheta))) - (Float32(pi) ^ Float32(-0.5))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + (c + ((single(pi) ^ single(-0.5)) / cosTheta))) - (single(pi) ^ single(-0.5)));
end

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. fma-define 97.9%

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

   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. associate--l- 98.5%

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

   6. sub-neg 98.5%

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

   7. neg-mul-1 98.5%

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

   8. distribute-lft-out 98.5%

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

   9. distribute-rgt-out 98.5%

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

   10. metadata-eval 98.5%

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

   11. distribute-lft-neg-out 98.5%

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

   12. distribute-rgt-neg-out 98.5%

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

   13. exp-prod 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0 95.4%

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

   1. +-commutative 95.4%

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

   2. mul-1-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in cosTheta around inf 95.4%

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

   1. *-un-lft-identity 95.4%

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

   2. associate--l+ 95.4%

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

   3. associate-*l/ 95.4%

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

   4. *-un-lft-identity 95.4%

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

   5. sqrt-div 95.4%

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

   6. metadata-eval 95.4%

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

   7. pow1/2 95.4%

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

   8. pow-flip 95.4%

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

   9. metadata-eval 95.4%

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

   10. sqrt-div 95.4%

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

   11. metadata-eval 95.4%

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

   12. pow1/2 95.4%

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

   13. pow-flip 95.4%

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

   14. metadata-eval 95.4%

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

10. Applied egg-rr 95.4%

    \[\leadsto \color{blue}{1 \cdot \frac{1}{1 + \left(\left(c + \frac{{\pi}^{-0.5}}{cosTheta}\right) - {\pi}^{-0.5}\right)}} \]
11. Step-by-step derivation

    1. *-lft-identity 95.4%

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

    2. associate-+r- 95.4%

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

12. Simplified 95.4%

    \[\leadsto \color{blue}{\frac{1}{\left(1 + \left(c + \frac{{\pi}^{-0.5}}{cosTheta}\right)\right) - {\pi}^{-0.5}}} \]
13. Add Preprocessing

Alternative 10: 94.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. fma-define 97.9%

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

   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. associate--l- 98.5%

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

   6. sub-neg 98.5%

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

   7. neg-mul-1 98.5%

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

   8. distribute-lft-out 98.5%

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

   9. distribute-rgt-out 98.5%

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

   10. metadata-eval 98.5%

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

   11. distribute-lft-neg-out 98.5%

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

   12. distribute-rgt-neg-out 98.5%

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

   13. exp-prod 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0 95.4%

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

   1. +-commutative 95.4%

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

   2. mul-1-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in cosTheta around inf 95.4%

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

9. Taylor expanded in c around 0 95.2%

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

    1. associate--l+ 95.2%

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

    2. sub-neg 95.2%

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

    3. mul-1-neg 95.2%

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

    4. +-commutative 95.2%

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

    5. distribute-rgt-out 95.1%

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

11. Simplified 95.1%

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

12. Final simplification 95.1%

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

Alternative 11: 92.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. fma-define 97.9%

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

   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. associate--l- 98.5%

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

   6. sub-neg 98.5%

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

   7. neg-mul-1 98.5%

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

   8. distribute-lft-out 98.5%

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

   9. distribute-rgt-out 98.5%

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

   10. metadata-eval 98.5%

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

   11. distribute-lft-neg-out 98.5%

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

   12. distribute-rgt-neg-out 98.5%

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

   13. exp-prod 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in cosTheta around 0 95.4%

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

   1. +-commutative 95.4%

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

   2. mul-1-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around inf 92.3%

   \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{\pi}} + cosTheta \cdot \color{blue}{c}}{cosTheta}} \]

9. Taylor expanded in cosTheta around 0 92.9%

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

    1. mul-1-neg 92.9%

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

    2. unsub-neg 92.9%

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

    3. *-commutative 92.9%

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

11. Simplified 92.9%

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

12. Final simplification 92.9%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} - c \cdot \left(\pi \cdot cosTheta\right)\right) \]
13. Add Preprocessing

Alternative 12: 92.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ 97.9%

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

   2. distribute-lft-neg-out 97.9%

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

   3. distribute-rgt-neg-out 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in cosTheta around 0 92.9%

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

6. Final simplification 92.9%

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

Alternative 13: 5.1% accurate, 107.3× speedup

Math

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

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 97.9%

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

   1. associate-+l+ 97.9%

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

   2. distribute-lft-neg-out 97.9%

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

   3. distribute-rgt-neg-out 97.9%

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

   4. exp-prod 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in c around inf 4.9%

   \[\leadsto \color{blue}{\frac{1}{c}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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))))))