Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.3%

Time: 14.8s

Alternatives: 12

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: 97.8% 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: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-define 98.1%

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

   3. associate-*l/ 98.6%

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

   4. *-lft-identity 98.6%

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

   5. sub-neg 98.6%

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

   6. associate--l+ 98.6%

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

   7. unsub-neg 98.6%

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

   8. neg-mul-1 98.6%

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

   9. neg-mul-1 98.6%

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

   10. distribute-rgt-out 98.6%

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

   11. metadata-eval 98.6%

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

   12. *-commutative 98.6%

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

   13. exp-prod 98.6%

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

3. Simplified 98.6%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(exp(Float32(-(cosTheta ^ Float32(2.0)))) / Float32(Float32(cosTheta * sqrt(Float32(pi))) / sqrt(fma(cosTheta, Float32(-2.0), Float32(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 98.0%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing
5. 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 98.6%

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

   3. inv-pow 98.6%

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

   4. +-commutative 98.6%

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

   5. fma-define 98.6%

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

6. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. unpow-1 98.6%

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

   3. associate-/r/ 98.5%

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

   4. pow-exp 98.6%

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

   5. distribute-lft-neg-out 98.6%

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

   6. unpow2 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

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

   2. fma-define 98.6%

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

   3. +-commutative 98.6%

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

   4. associate-*l/ 98.5%

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

   5. neg-mul-1 98.5%

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

   6. *-lft-identity 98.5%

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

   7. neg-mul-1 98.5%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

10. Simplified 98.6%

    \[\leadsto \color{blue}{\frac{1}{1 + \left(c + \frac{e^{-{cosTheta}^{2}}}{\frac{cosTheta \cdot \sqrt{\pi}}{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}\right)}} \]
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 98.0%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing
5. 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 98.6%

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

   3. inv-pow 98.6%

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

   4. +-commutative 98.6%

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

   5. fma-define 98.6%

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

6. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. unpow-1 98.6%

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

   3. associate-/r/ 98.5%

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

   4. pow-exp 98.6%

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

   5. distribute-lft-neg-out 98.6%

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

   6. unpow2 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

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

   2. fma-define 98.6%

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

   3. +-commutative 98.6%

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

   4. associate-*l/ 98.5%

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

   5. neg-mul-1 98.5%

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

   6. *-lft-identity 98.5%

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

   7. neg-mul-1 98.5%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

10. Simplified 98.6%

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

11. Taylor expanded in c around 0 98.2%

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

    1. unpow2 98.2%

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

13. Applied egg-rr 98.2%

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

14. Final simplification 98.2%

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

Alternative 4: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.0%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta around 0 96.3%

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

   1. associate-*r* 96.3%

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

   2. mul-1-neg 96.3%

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

   3. mul-1-neg 96.3%

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

7. Simplified 96.3%

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

8. Final simplification 96.3%

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

Alternative 5: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.0%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta around 0 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

   3. associate-*r* 96.3%

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

   4. associate-+r+ 96.3%

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

   5. mul-1-neg 96.3%

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

   6. unsub-neg 96.3%

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

7. Simplified 96.3%

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

8. Final simplification 96.3%

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

Alternative 6: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta around 0 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

   3. associate-*r* 96.3%

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

   4. associate-+r+ 96.3%

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

   5. mul-1-neg 96.3%

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

   6. unsub-neg 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in c around 0 96.3%

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

9. Final simplification 96.3%

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

Alternative 7: 95.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.0%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing
5. 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 98.6%

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

   3. inv-pow 98.6%

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

   4. +-commutative 98.6%

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

   5. fma-define 98.6%

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

6. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. unpow-1 98.6%

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

   3. associate-/r/ 98.5%

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

   4. pow-exp 98.6%

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

   5. distribute-lft-neg-out 98.6%

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

   6. unpow2 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-lft-identity 98.6%

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

   2. fma-define 98.6%

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

   3. +-commutative 98.6%

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

   4. associate-*l/ 98.5%

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

   5. neg-mul-1 98.5%

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

   6. *-lft-identity 98.5%

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

   7. neg-mul-1 98.5%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

10. Simplified 98.6%

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

11. Taylor expanded in c around 0 98.2%

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

12. Taylor expanded in cosTheta around 0 96.0%

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

13. Final simplification 96.0%

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

Alternative 8: 95.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

7. Simplified 95.7%

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

8. Taylor expanded in cosTheta around inf 95.8%

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

   1. associate--l+ 95.8%

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

   2. sub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. +-commutative 95.8%

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

   5. distribute-rgt-out 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 9: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta around 0 95.7%

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

   1. mul-1-neg 95.7%

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

7. Simplified 95.7%

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

8. Taylor expanded in c around 0 95.7%

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

   1. associate--l+ 95.7%

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

   2. sub-neg 95.7%

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

   3. mul-1-neg 95.7%

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

   4. +-commutative 95.7%

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

   5. distribute-rgt-out 95.7%

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

10. Simplified 95.7%

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

11. Final simplification 95.7%

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

Alternative 10: 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 98.0%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}, {\left(e^{-cosTheta}\right)}^{cosTheta}, c\right)}} \]
4. Add Preprocessing
5. 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 98.6%

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

   3. inv-pow 98.6%

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

   4. +-commutative 98.6%

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

   5. fma-define 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in cosTheta around 0 93.3%

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

Alternative 11: 10.7% accurate, 107.3× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 (- 1.0 c))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.0%

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

3. Simplified 98.6%

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

5. Taylor expanded in c around inf 10.7%

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

6. Taylor expanded in c around 0 10.7%

   \[\leadsto \color{blue}{1 + -1 \cdot c} \]
7. Step-by-step derivation

   1. mul-1-neg 10.7%

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

   2. unsub-neg 10.7%

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

8. Simplified 10.7%

   \[\leadsto \color{blue}{1 - c} \]
9. Add Preprocessing

Alternative 12: 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 98.0%

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

3. Simplified 98.6%

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

5. Taylor expanded in c around inf 10.7%

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

6. Taylor expanded in c around 0 10.7%

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

Reproduce

herbie shell --seed 2024150 
(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))))))