Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.4%

Time: 13.6s

Alternatives: 13

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% 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 97.7%

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

   1. +-commutative 97.7%

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

   2. fma-define 97.7%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate--l- 98.4%

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

   6. sub-neg 98.4%

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

   7. neg-mul-1 98.4%

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

   8. distribute-lft-out 98.4%

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

   9. distribute-rgt-out 98.4%

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

   10. metadata-eval 98.4%

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

   11. distribute-lft-neg-out 98.4%

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

   12. distribute-rgt-neg-out 98.4%

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

   13. exp-prod 98.4%

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

3. Simplified 98.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. add-cube-cbrt 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. pow3 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{{\left(\sqrt[3]{\pi}\right)}^{3}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt{{\left(\sqrt[3]{\pi}\right)}^{3}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. associate--l- 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\color{blue}{1 - \left(cosTheta + cosTheta\right)}}}{cosTheta}}{\sqrt{{\left(\sqrt[3]{\pi}\right)}^{3}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. rem-cube-cbrt 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. add-cube-cbrt 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. pow3 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{{\left(\sqrt[3]{\pi}\right)}^{3}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt{{\left(\sqrt[3]{\pi}\right)}^{3}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. associate--l- 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\color{blue}{1 - \left(cosTheta + cosTheta\right)}}}{cosTheta}}{\sqrt{{\left(\sqrt[3]{\pi}\right)}^{3}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. rem-cube-cbrt 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in c around 0 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 98.0% accurate, 1.5× speedup

Math

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

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(-Float32(cosTheta * 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around 0 97.9%

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

   1. unpow2 97.9%

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

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

Alternative 5: 97.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around 0 97.9%

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

6. Taylor expanded in cosTheta around 0 96.8%

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

   1. neg-mul-1 96.8%

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

   2. unsub-neg 96.8%

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

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

Alternative 6: 96.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + ((cosTheta * single(pi)) * (single(-1.0) + ((single(pi) ^ single(-0.5)) - c))));
end

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

   3. associate-*r* 96.4%

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

   4. associate-+r+ 96.4%

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

   5. mul-1-neg 96.4%

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

   6. unsub-neg 96.4%

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

7. Simplified 96.4%

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

   1. sqrt-div 96.4%

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

   2. metadata-eval 96.4%

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

   3. associate--l+ 96.4%

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

   4. pow1/2 96.4%

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

   5. pow-flip 96.4%

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

   6. metadata-eval 96.4%

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

9. Applied egg-rr 96.4%

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

10. Final simplification 96.4%

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

Alternative 7: 95.9% 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

   3. associate-*r* 96.4%

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

   4. associate-+r+ 96.4%

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

   5. mul-1-neg 96.4%

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

   6. unsub-neg 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in c around 0 96.2%

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

9. Final simplification 96.2%

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

Alternative 8: 95.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around 0 97.9%

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

6. Taylor expanded in cosTheta around 0 95.6%

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

7. Final simplification 95.6%

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

Alternative 9: 95.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 95.4%

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

   1. mul-1-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in cosTheta around inf 95.4%

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

   1. associate--l+ 95.4%

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

   2. associate--l+ 95.4%

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

   3. sub-neg 95.4%

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

   4. mul-1-neg 95.4%

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

   5. +-commutative 95.4%

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

   6. distribute-rgt-out 95.4%

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

   7. unpow1/2 95.4%

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

   8. rem-exp-log 95.4%

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

   9. rec-exp 95.4%

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

   10. exp-prod 95.4%

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

   11. distribute-lft-neg-out 95.4%

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

   12. rec-exp 95.4%

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

   13. exp-to-pow 95.4%

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

   14. unpow1/2 95.4%

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

10. Simplified 95.4%

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

Alternative 10: 95.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 95.4%

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

   1. mul-1-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around 0 95.2%

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

   1. associate--l+ 95.2%

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

   2. sub-neg 95.2%

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

   3. mul-1-neg 95.2%

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

   4. +-commutative 95.2%

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

   5. distribute-rgt-out 95.2%

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

   6. unpow1/2 95.2%

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

   7. rem-exp-log 95.2%

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

   8. rec-exp 95.2%

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

   9. exp-prod 95.2%

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

   10. distribute-lft-neg-out 95.2%

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

   11. rec-exp 95.2%

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

   12. exp-to-pow 95.2%

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

   13. unpow1/2 95.2%

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

10. Simplified 95.2%

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

Alternative 11: 93.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

   3. associate-*r* 96.4%

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

   4. associate-+r+ 96.4%

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

   5. mul-1-neg 96.4%

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

   6. unsub-neg 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in c around inf 93.6%

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

9. Final simplification 93.6%

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

Alternative 12: 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 95.4%

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

   1. mul-1-neg 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in cosTheta around 0 93.6%

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

Alternative 13: 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 97.7%

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

   1. associate-+l+ 97.7%

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

   2. +-commutative 97.7%

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

   3. fma-define 97.7%

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

   4. times-frac 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around inf 10.5%

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

6. Taylor expanded in c around 0 10.5%

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

Reproduce

herbie shell --seed 2024141 
(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))))))