Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 14.8s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((single(1.0) / ((sqrt(single(pi)) * cosTheta) / sqrt(((single(1.0) - cosTheta) - cosTheta)))) * 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. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. clear-num 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (sqrtf(((1.0f - cosTheta) - cosTheta)) / ((sqrtf(((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(Float32(Float32(1.0) - cosTheta) - cosTheta)) / Float32(Float32(sqrt(Float32(pi)) * cosTheta) * exp(Float32(cosTheta * cosTheta))))))
end

MATLAB

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

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

   3. exp-prod 98.0%

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

3. Simplified 98.0%

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

   1. associate-*r* 98.0%

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

   2. pow-neg 98.0%

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

   3. pow-exp 98.0%

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

   4. div-inv 98.0%

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

   5. frac-times 98.6%

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

   6. *-un-lft-identity 98.6%

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

   7. associate-/l/ 98.6%

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

5. Applied egg-rr 98.6%

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

6. Final simplification 98.6%

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

Alternative 3: 97.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (((single(1.0) - cosTheta) + ((cosTheta * cosTheta) * single(-0.5))) / (sqrt(single(pi)) * (cosTheta * 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. 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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 97.9%

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

   1. neg-mul-1 97.9%

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

   2. associate-+r+ 97.9%

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

   3. sub-neg 97.9%

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

   4. *-commutative 97.9%

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

   5. unpow2 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (sqrtf(((1.0f + (cosTheta * -2.0f)) / ((float) M_PI))) * (1.0f / (cosTheta * expf((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(Float32(1.0) / Float32(cosTheta * exp(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))) * (single(1.0) / (cosTheta * 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. 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. *-commutative 98.0%

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

   3. exp-prod 98.0%

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

3. Simplified 98.0%

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

   1. associate-*r* 98.0%

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

   2. pow-neg 98.0%

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

   3. pow-exp 98.0%

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

   4. div-inv 98.0%

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

   5. frac-times 98.6%

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

   6. *-un-lft-identity 98.6%

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

   7. associate-/l/ 98.6%

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

5. Applied egg-rr 98.6%

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

6. Taylor expanded in c around 0 97.8%

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

   1. cancel-sign-sub-inv 97.8%

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

   2. metadata-eval 97.8%

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

   3. *-commutative 97.8%

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

   4. unpow2 97.8%

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

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 5: 96.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   4. *-lft-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 97.1%

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

   1. neg-mul-1 97.1%

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

   2. sub-neg 97.1%

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

6. Simplified 97.1%

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

7. Final simplification 97.1%

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

Alternative 6: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (c + (single(1.0) + ((single(-1.0) + (single(1.0) / 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%

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

   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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 96.2%

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

   1. distribute-rgt-out 96.2%

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

6. Simplified 96.2%

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

   1. *-un-lft-identity 96.2%

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

   2. sqrt-div 96.2%

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

   3. metadata-eval 96.2%

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

8. Applied egg-rr 96.2%

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

   1. *-lft-identity 96.2%

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

   2. +-commutative 96.2%

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

   3. associate-+l+ 96.2%

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

   4. associate-*l/ 96.6%

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

   5. *-lft-identity 96.6%

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

10. Simplified 96.6%

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

11. Final simplification 96.6%

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

Alternative 7: 95.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(-1.0) + (single(1.0) / 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%

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

   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/ 98.5%

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

   4. *-lft-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 96.2%

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

   1. distribute-rgt-out 96.2%

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

6. Simplified 96.2%

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

7. Taylor expanded in c around 0 96.1%

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

   1. sub-neg 96.1%

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

   2. metadata-eval 96.1%

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

   3. +-commutative 96.1%

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

   4. *-commutative 96.1%

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

   5. sqrt-div 96.1%

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

   6. metadata-eval 96.1%

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

   7. associate-*l/ 96.5%

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

   8. *-un-lft-identity 96.5%

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

   9. +-commutative 96.5%

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

9. Applied egg-rr 96.5%

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

10. Final simplification 96.5%

    \[\leadsto \frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \]

Alternative 8: 93.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. clear-num 98.6%

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

3. Applied egg-rr 98.6%

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

4. Taylor expanded in cosTheta around 0 94.3%

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

5. Final simplification 94.3%

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

Alternative 9: 10.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   4. *-lft-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 96.8%

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

   1. fma-def 96.9%

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

   2. associate-*r* 96.9%

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

   3. mul-1-neg 96.9%

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

   4. unpow2 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-+r+ 96.9%

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

   7. +-commutative 96.9%

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

   8. mul-1-neg 96.9%

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

6. Simplified 96.9%

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

7. Taylor expanded in c around inf 10.6%

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

   1. mul-1-neg 10.6%

      \[\leadsto \color{blue}{-c \cdot \left({cosTheta}^{2} \cdot \pi\right)} \]

   2. distribute-rgt-neg-in 10.6%

      \[\leadsto \color{blue}{c \cdot \left(-{cosTheta}^{2} \cdot \pi\right)} \]

   3. *-commutative 10.6%

      \[\leadsto c \cdot \left(-\color{blue}{\pi \cdot {cosTheta}^{2}}\right) \]

   4. unpow2 10.6%

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

9. Simplified 10.6%

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

10. Final simplification 10.6%

    \[\leadsto c \cdot \left(-\pi \cdot \left(cosTheta \cdot cosTheta\right)\right) \]

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 97.7%

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

3. Taylor expanded in cosTheta around inf 10.4%

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

4. Final simplification 10.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(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))))))