Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 15.7s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (expf((cosTheta * -cosTheta)) / ((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(-cosTheta))) / 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 98.7%

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

   1. associate-/l/ 98.5%

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

   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. fma-undefine 98.6%

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

   2. unpow-1 98.6%

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

   3. associate-/r/ 98.7%

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

   4. pow-exp 98.7%

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

   5. distribute-lft-neg-out 98.7%

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

   6. pow2 98.7%

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

8. Applied egg-rr 98.7%

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

   1. +-commutative 98.7%

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

   2. associate-*l/ 98.7%

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

   3. neg-mul-1 98.7%

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

   4. *-lft-identity 98.7%

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

   5. neg-mul-1 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-commutative 98.7%

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

10. Simplified 98.7%

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

    1. unpow2 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((expf(-powf(cosTheta, 2.0f)) / 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(c + Float32(Float32(exp(Float32(-(cosTheta ^ Float32(2.0)))) / 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) + (c + ((exp(-(cosTheta ^ single(2.0))) / 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. 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 98.7%

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

5. Taylor expanded in c around 0 98.0%

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

6. Final simplification 98.0%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((expf(-powf(cosTheta, 2.0f)) / 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(exp(Float32(-(cosTheta ^ Float32(2.0)))) / 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) + ((exp(-(cosTheta ^ single(2.0))) / 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 97.8%

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

4. Final simplification 97.8%

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

Alternative 4: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c - (exp((cosTheta * -cosTheta)) / ((cosTheta * sqrt(single(pi))) / (single(-1.0) - (cosTheta * (single(-1.0) + (cosTheta * single(-0.5)))))))));
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 98.7%

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

   1. associate-/l/ 98.5%

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

   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. fma-undefine 98.6%

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

   2. unpow-1 98.6%

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

   3. associate-/r/ 98.7%

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

   4. pow-exp 98.7%

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

   5. distribute-lft-neg-out 98.7%

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

   6. pow2 98.7%

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

8. Applied egg-rr 98.7%

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

   1. +-commutative 98.7%

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

   2. associate-*l/ 98.7%

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

   3. neg-mul-1 98.7%

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

   4. *-lft-identity 98.7%

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

   5. neg-mul-1 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-commutative 98.7%

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

10. Simplified 98.7%

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

    1. unpow2 98.7%

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

12. Applied egg-rr 98.7%

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

13. Taylor expanded in cosTheta around 0 97.5%

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

14. Final simplification 97.5%

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

Alternative 5: 96.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

3. Simplified 98.7%

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

   1. associate-/l/ 98.5%

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

   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. fma-undefine 98.6%

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

   2. unpow-1 98.6%

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

   3. associate-/r/ 98.7%

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

   4. pow-exp 98.7%

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

   5. distribute-lft-neg-out 98.7%

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

   6. pow2 98.7%

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

8. Applied egg-rr 98.7%

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

   1. +-commutative 98.7%

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

   2. associate-*l/ 98.7%

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

   3. neg-mul-1 98.7%

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

   4. *-lft-identity 98.7%

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

   5. neg-mul-1 98.7%

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

   6. associate-*l/ 98.7%

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

   7. *-commutative 98.7%

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

10. Simplified 98.7%

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

    1. unpow2 98.7%

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

12. Applied egg-rr 98.7%

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

13. Taylor expanded in cosTheta around 0 96.8%

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

    1. mul-1-neg 96.8%

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

    2. sub-neg 96.8%

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

15. Simplified 96.8%

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

16. Final simplification 96.8%

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

Alternative 6: 95.7% 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 98.7%

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

5. Taylor expanded in cosTheta around 0 96.6%

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

   1. associate-*r* 96.6%

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

   2. mul-1-neg 96.6%

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

   3. mul-1-neg 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 7: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + ((cosTheta * single(pi)) * ((single(pi) ^ single(-0.5)) + (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 98.7%

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

5. Taylor expanded in cosTheta around 0 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. associate-*r* 96.6%

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

   4. associate-+r+ 96.6%

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

   5. mul-1-neg 96.6%

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

   6. unsub-neg 96.6%

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

7. Simplified 96.6%

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

   1. associate--l+ 96.6%

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

   2. pow1/2 96.6%

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

   3. inv-pow 96.6%

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

   4. pow-pow 96.6%

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

   5. metadata-eval 96.6%

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

9. Applied egg-rr 96.6%

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

    1. associate-+r- 96.6%

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

11. Simplified 96.6%

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

12. Final simplification 96.6%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + \left(cosTheta \cdot \pi\right) \cdot \left({\pi}^{-0.5} + \left(-1 - c\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 95.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(Float32(cosTheta * Float32(pi)) * Float32(Float32(-1.0) + 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) + 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 98.7%

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

5. Taylor expanded in cosTheta around 0 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

   3. associate-*r* 96.6%

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

   4. associate-+r+ 96.6%

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

   5. mul-1-neg 96.6%

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

   6. unsub-neg 96.6%

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

7. Simplified 96.6%

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

8. Taylor expanded in c around 0 96.5%

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

9. Final simplification 96.5%

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

Alternative 9: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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(-1.0) + Float32(Float32(1.0) / cosTheta))))))
end

MATLAB

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

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta around 0 95.9%

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

   1. mul-1-neg 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in cosTheta around inf 95.9%

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

   1. associate--l+ 95.9%

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

   2. sub-neg 95.9%

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

   3. mul-1-neg 95.9%

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

   4. +-commutative 95.9%

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

   5. distribute-rgt-out 95.9%

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

10. Simplified 95.9%

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

Alternative 10: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(sqrt(Float32(Float32(1.0) / Float32(pi))) * Float32(Float32(-1.0) + 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))) * (single(-1.0) + (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 98.0%

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta around 0 95.9%

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

   1. mul-1-neg 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in c around 0 95.8%

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

   1. associate--l+ 95.8%

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

   2. sub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. +-commutative 95.8%

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

   5. distribute-rgt-out 95.8%

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

10. Simplified 95.8%

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

Alternative 11: 92.8% 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 98.7%

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

   1. associate-/l/ 98.5%

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

   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 94.5%

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

Alternative 12: 10.8% 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 98.7%

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

5. Taylor expanded in c around inf 10.4%

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

6. Taylor expanded in c around 0 10.4%

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

   1. mul-1-neg 10.4%

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

   2. unsub-neg 10.4%

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

8. Simplified 10.4%

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

9. Taylor expanded in c around 0 10.4%

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

Reproduce

herbie shell --seed 2024186 
(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))))))