Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 11.4s

Alternatives: 10

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   2. associate-+l+ 98.0%

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

   3. distribute-lft-neg-out 98.0%

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

   4. exp-neg 98.0%

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

   5. associate-*r/ 98.0%

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

   6. associate-/l* 98.0%

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

   7. /-rgt-identity 98.0%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 98.0%

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

   2. *-lft-identity 98.0%

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

   3. associate-*r* 98.0%

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

   4. *-lft-identity 98.0%

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

   5. associate--l- 98.0%

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

   6. *-commutative 98.0%

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

3. Simplified 98.0%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

5. Applied egg-rr 98.6%

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

6. Final simplification 98.6%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) / Float32(cosTheta * exp(Float32(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. associate-*r/ 98.1%

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

   2. distribute-lft-neg-out 98.1%

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

   3. exp-neg 98.1%

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

   4. times-frac 98.1%

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

   5. *-commutative 98.1%

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

   6. times-frac 98.0%

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

   7. associate-*l/ 98.0%

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

3. Simplified 98.6%

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

4. Taylor expanded in c around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. +-commutative 98.2%

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

   3. cancel-sign-sub-inv 98.2%

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

   4. metadata-eval 98.2%

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

   5. associate-+l+ 98.2%

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

6. Simplified 98.3%

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

7. Final simplification 98.3%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))) * (exp((cosTheta * -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. Taylor expanded in c around 0 97.6%

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

   1. add-log-exp 97.4%

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

   2. cancel-sign-sub-inv 97.4%

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

   3. metadata-eval 97.4%

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

4. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.5%

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

   2. mul-1-neg 97.5%

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

   3. pow2 97.5%

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

   4. add-log-exp 97.7%

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

   5. *-commutative 97.7%

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

6. Applied egg-rr 97.7%

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

   1. *-commutative 97.7%

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

   2. associate-*r/ 97.6%

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

   3. distribute-rgt-neg-in 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in c around 0 97.6%

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

   1. add-log-exp 97.4%

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

   2. cancel-sign-sub-inv 97.4%

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

   3. metadata-eval 97.4%

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

4. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.5%

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

   2. mul-1-neg 97.5%

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

   3. pow2 97.5%

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

   4. add-log-exp 97.7%

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

   5. *-commutative 97.7%

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

6. Applied egg-rr 97.7%

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

7. Final simplification 97.7%

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

Alternative 6: 97.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((((cosTheta * -1.5f) + (((cosTheta * cosTheta) * 0.5f) + (1.0f / cosTheta))) + -1.0f) / sqrtf(((float) M_PI))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32(Float32(cosTheta * cosTheta) * Float32(0.5)) + Float32(Float32(1.0) / cosTheta))) + Float32(-1.0)) / sqrt(Float32(pi)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((((cosTheta * single(-1.5)) + (((cosTheta * cosTheta) * single(0.5)) + (single(1.0) / cosTheta))) + single(-1.0)) / sqrt(single(pi))));
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 98.0%

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

   2. *-lft-identity 98.0%

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

   3. associate-*r* 98.0%

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

   4. *-lft-identity 98.0%

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

   5. associate--l- 98.0%

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

   6. *-commutative 98.0%

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

3. Simplified 98.0%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

5. Applied egg-rr 98.6%

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

6. Taylor expanded in cosTheta around 0 97.4%

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

   1. pow1 97.4%

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

   2. pow2 97.4%

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

8. Applied egg-rr 97.4%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{\left(0.5 \cdot \left(cosTheta \cdot cosTheta\right)\right)}^{1}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}} \]
9. Step-by-step derivation

   1. unpow1 97.4%

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

   2. unpow2 97.4%

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

   3. *-commutative 97.4%

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

   4. unpow2 97.4%

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

10. Simplified 97.4%

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

11. Final simplification 97.4%

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

Alternative 7: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((((cosTheta * -1.5f) + (1.0f / cosTheta)) + -1.0f) / sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32(1.0) / cosTheta)) + Float32(-1.0)) / sqrt(Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((((cosTheta * single(-1.5)) + (single(1.0) / cosTheta)) + single(-1.0)) / sqrt(single(pi)))));
end

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 96.9%

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

5. Final simplification 96.9%

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

Alternative 8: 95.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 98.0%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 95.8%

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

5. Final simplification 95.8%

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

Alternative 9: 93.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 98.0%

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

   2. *-lft-identity 98.0%

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

   3. associate-*r* 98.0%

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

   4. *-lft-identity 98.0%

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

   5. associate--l- 98.0%

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

   6. *-commutative 98.0%

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

3. Simplified 98.0%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

5. Applied egg-rr 98.6%

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

6. Taylor expanded in cosTheta around 0 93.6%

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

7. Final simplification 93.6%

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

Alternative 10: 10.7% accurate, 421.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

float code(float cosTheta, float c) {
	return 1.0f;
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0
end function

Julia

function code(cosTheta, c)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0);
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in c around 0 97.6%

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

3. Taylor expanded in cosTheta around inf 10.6%

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

4. Final simplification 10.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023182 
(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))))))