Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 17.8s

Alternatives: 11

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% 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-define 98.0%

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

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. Add Preprocessing

5. Final simplification 98.6%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((exp(-(cosTheta ^ single(2.0))) / cosTheta) * sqrt((cosTheta * ((single(1.0) / (cosTheta * single(pi))) - (single(2.0) / 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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 c around 0 98.2%

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

6. Taylor expanded in cosTheta around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((sqrt((cosTheta * ((single(1.0) / (cosTheta * single(pi))) - (single(2.0) / single(pi))))) * exp(-(cosTheta ^ single(2.0)))) / 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. +-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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 c around 0 98.2%

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

6. Taylor expanded in cosTheta around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

8. Simplified 98.2%

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

   1. associate-*l/ 98.2%

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

   2. mul-1-neg 98.2%

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

   3. *-commutative 98.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((exp(-(cosTheta ^ single(2.0))) / cosTheta) * sqrt(((single(1.0) + (cosTheta * single(-2.0))) / 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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 c around 0 98.2%

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

6. Final simplification 98.2%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Add Preprocessing

3. Taylor expanded in c around 0 98.0%

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

4. Final simplification 98.0%

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

Alternative 6: 95.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + (single(pi) * (cosTheta * (sqrt((single(1.0) / single(pi))) + (single(-1.0) - 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-define 98.0%

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

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. Add Preprocessing

5. Taylor expanded in cosTheta around 0 97.3%

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

   1. mul-1-neg 97.3%

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

   2. unsub-neg 97.3%

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

   3. associate-*r* 97.3%

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

   4. *-commutative 97.3%

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

   5. associate-*l* 97.3%

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

   6. associate-+r+ 97.3%

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

   7. mul-1-neg 97.3%

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

   8. unsub-neg 97.3%

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 7: 95.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 96.5%

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

   1. +-commutative 96.5%

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

   2. mul-1-neg 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in cosTheta around 0 97.3%

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

   1. associate-*r* 97.3%

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

   2. mul-1-neg 97.3%

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

   3. sub-neg 97.3%

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

   4. mul-1-neg 97.3%

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

   5. associate-+r+ 97.3%

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

   6. distribute-lft-neg-in 97.3%

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

   7. unsub-neg 97.3%

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

   8. associate-*r* 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-+r+ 97.3%

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

   11. mul-1-neg 97.3%

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

   12. sub-neg 97.3%

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

10. Simplified 97.3%

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

11. Taylor expanded in c around 0 97.2%

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

12. Final simplification 97.2%

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

Alternative 8: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((sqrt((single(1.0) / single(pi))) * (single(1.0) - cosTheta)) / 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. +-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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 c around 0 98.2%

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

6. Taylor expanded in cosTheta around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

8. Simplified 98.2%

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

   1. associate-*l/ 98.2%

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

   2. mul-1-neg 98.2%

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

   3. *-commutative 98.2%

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

10. Applied egg-rr 98.2%

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

11. Taylor expanded in cosTheta around 0 96.5%

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

    1. associate-*r* 96.5%

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

    2. neg-mul-1 96.5%

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

    3. distribute-rgt1-in 96.5%

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

13. Simplified 96.5%

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

14. Final simplification 96.5%

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

Alternative 9: 92.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) - (c * (cosTheta * 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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 96.5%

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

   1. +-commutative 96.5%

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

   2. mul-1-neg 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in cosTheta around 0 97.3%

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

   1. associate-*r* 97.3%

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

   2. mul-1-neg 97.3%

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

   3. sub-neg 97.3%

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

   4. mul-1-neg 97.3%

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

   5. associate-+r+ 97.3%

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

   6. distribute-lft-neg-in 97.3%

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

   7. unsub-neg 97.3%

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

   8. associate-*r* 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-+r+ 97.3%

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

   11. mul-1-neg 97.3%

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

   12. sub-neg 97.3%

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

10. Simplified 97.3%

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

11. Taylor expanded in c around inf 94.4%

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

12. Final simplification 94.4%

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

Alternative 10: 92.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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-define 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   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 c around 0 98.2%

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

6. Taylor expanded in cosTheta around 0 94.4%

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

Alternative 11: 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 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-define 98.0%

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

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. Add Preprocessing

5. Taylor expanded in c around inf 5.0%

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

Reproduce

herbie shell --seed 2024086 
(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))))))