Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 9.2s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate--r+ 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative 97.6%

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

   2. associate-+l+ 97.6%

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

   3. distribute-lft-neg-out 97.6%

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

   4. exp-neg 97.6%

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

   5. associate-*r/ 97.7%

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

   6. associate-/l* 97.7%

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

   7. /-rgt-identity 97.7%

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

3. Simplified 98.5%

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

4. Taylor expanded in c around 0 97.2%

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

   1. un-div-inv 97.6%

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

   2. *-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. pow2 97.6%

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

6. Applied egg-rr 97.6%

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

7. Final simplification 97.6%

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

Alternative 3: 96.8% accurate, 1.9× speedup

Math

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

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(Float32(1.0) + 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 97.6%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. associate-*r* 97.6%

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

   4. *-lft-identity 97.6%

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

   5. associate--l- 97.6%

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

   6. *-commutative 97.6%

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

3. Simplified 97.6%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate-*l/ 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in cosTheta around 0 96.5%

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

7. Final simplification 96.5%

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

Alternative 4: 95.4% accurate, 2.0× speedup

Math

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

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(Float32(1.0) + 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 97.6%

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

   1. associate-*l* 97.6%

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

   2. *-lft-identity 97.6%

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

   3. associate-*r* 97.6%

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

   4. *-lft-identity 97.6%

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

   5. associate--l- 97.6%

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

   6. *-commutative 97.6%

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

3. Simplified 97.6%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. associate-*l/ 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in cosTheta around 0 95.0%

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

7. Final simplification 95.0%

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

Alternative 5: 92.9% 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 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate--r+ 98.6%

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

3. Applied egg-rr 98.6%

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

4. Taylor expanded in cosTheta around 0 92.6%

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

5. Final simplification 92.6%

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

Alternative 6: 10.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate--r+ 98.6%

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

3. Applied egg-rr 98.6%

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

4. Taylor expanded in cosTheta around 0 95.3%

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

   1. fma-def 95.4%

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

   2. associate-*r* 95.4%

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

   3. mul-1-neg 95.4%

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

   4. unpow2 95.4%

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

   5. *-commutative 95.4%

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

   6. mul-1-neg 95.4%

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

   7. sub-neg 95.4%

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

6. Simplified 95.4%

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

7. Taylor expanded in c around inf 10.8%

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

   1. associate-*r* 10.8%

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

   2. mul-1-neg 10.8%

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

   3. unpow2 10.8%

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

9. Simplified 10.8%

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

10. Final simplification 10.8%

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

Alternative 7: 10.8% 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 97.6%

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

   1. +-commutative 97.6%

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

   2. associate-+l+ 97.6%

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

   3. distribute-lft-neg-out 97.6%

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

   4. exp-neg 97.6%

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

   5. associate-*r/ 97.7%

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

   6. associate-/l* 97.7%

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

   7. /-rgt-identity 97.7%

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

3. Simplified 98.5%

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

4. Taylor expanded in c around 0 97.2%

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

   1. +-commutative 97.2%

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

   2. associate-*r/ 97.6%

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

   3. metadata-eval 97.6%

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

   4. cancel-sign-sub-inv 97.6%

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

   5. *-rgt-identity 97.6%

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

   6. cancel-sign-sub-inv 97.6%

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

   7. metadata-eval 97.6%

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

   8. +-commutative 97.6%

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

   9. *-commutative 97.6%

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

   10. fma-udef 97.6%

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

   11. *-commutative 97.6%

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

   12. unpow2 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in cosTheta around inf 10.8%

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

8. Final simplification 10.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023174 
(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))))))