Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.6%

Time: 12.9s

Alternatives: 10

Speedup: 1.5×

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: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 97.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Taylor expanded in c around 0 98.2%

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

   1. unpow2 98.2%

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

7. Applied egg-rr 98.2%

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

   1. metadata-eval 98.2%

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

   2. cancel-sign-sub-inv 98.2%

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

   3. div-sub 98.2%

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

   4. *-commutative 98.2%

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

9. Applied egg-rr 98.2%

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

10. Final simplification 98.2%

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

Alternative 3: 97.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Taylor expanded in c around 0 98.2%

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

   1. unpow2 98.2%

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

7. Applied egg-rr 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi))) * ((single(1.0) - (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. Step-by-step derivation

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Taylor expanded in c around 0 98.2%

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

6. Taylor expanded in cosTheta around 0 97.5%

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

   1. neg-mul-1 97.5%

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

   2. sub-neg 97.5%

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

8. Simplified 97.5%

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

   1. unpow2 98.2%

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

10. Applied egg-rr 97.5%

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

11. Final simplification 97.5%

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

Alternative 5: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Taylor expanded in c around 0 98.2%

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

6. Taylor expanded in cosTheta around 0 97.5%

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

   1. neg-mul-1 97.5%

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

   2. sub-neg 97.5%

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

8. Simplified 97.5%

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

   1. div-sub 97.5%

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

   2. pow1 97.5%

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

   3. pow-div 97.5%

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

   4. metadata-eval 97.5%

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

   5. pow1 97.5%

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

10. Applied egg-rr 97.5%

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

11. Final simplification 97.5%

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

Alternative 6: 95.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Taylor expanded in c around 0 98.2%

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

6. Taylor expanded in cosTheta around 0 95.6%

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

7. Final simplification 95.6%

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

Alternative 7: 95.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0 97.3%

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

4. Taylor expanded in cosTheta around 0 95.2%

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

   1. div-sub 95.2%

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

   2. *-commutative 95.2%

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

6. Applied egg-rr 95.2%

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

7. Final simplification 95.2%

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

Alternative 8: 95.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in c around 0 97.3%

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

4. Taylor expanded in cosTheta around 0 95.2%

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

5. Final simplification 95.2%

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

Alternative 9: 93.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

   1. associate-/l/ 98.6%

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

   2. clear-num 98.7%

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

   3. inv-pow 98.7%

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

   4. +-commutative 98.7%

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

   5. fma-define 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in cosTheta around 0 93.5%

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

Alternative 10: 10.7% accurate, 322.0× 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. Step-by-step derivation

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

3. Simplified 98.8%

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

5. Taylor expanded in c around inf 10.5%

   \[\leadsto \frac{1}{1 + \color{blue}{c}} \]

6. Taylor expanded in c around 0 10.5%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))))))