Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.4%

Time: 19.6s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% 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 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-define 98.1%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. associate--l- 98.7%

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

   6. sub-neg 98.7%

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

   7. neg-mul-1 98.7%

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

   8. distribute-lft-out 98.7%

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

   9. distribute-rgt-out 98.7%

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

   10. metadata-eval 98.7%

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

   11. distribute-lft-neg-out 98.7%

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

   12. distribute-rgt-neg-out 98.7%

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

   13. exp-prod 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

   4. times-frac 98.7%

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

   5. *-lft-identity 98.7%

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

   6. associate--l- 98.7%

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

   7. sub-neg 98.7%

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

   8. neg-mul-1 98.7%

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

   9. distribute-lft-out 98.7%

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

   10. distribute-rgt-out 98.7%

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

   11. metadata-eval 98.7%

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

   12. exp-prod 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((powf(((float) M_PI), -0.5f) * 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(pi) ^ Float32(-0.5)) * sqrt(Float32(Float32(1.0) - Float32(cosTheta + cosTheta)))) / cosTheta) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. add-cube-cbrt 98.0%

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

   2. cbrt-unprod 98.0%

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

   3. frac-times 97.9%

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

   4. metadata-eval 97.9%

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

   5. add-sqr-sqrt 97.9%

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

   6. inv-pow 97.9%

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

   7. sqrt-pow2 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. metadata-eval 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{\color{blue}{-0.5}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right) \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. cbrt-unprod 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\sqrt[3]{\frac{1}{\pi} \cdot {\pi}^{-0.5}}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. inv-pow 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\color{blue}{{\pi}^{-1}} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. metadata-eval 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{{\pi}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. pow-prod-up 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. add-cbrt-cube 98.2%

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

   7. associate--l- 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 4: 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. +-commutative 98.0%

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

   2. fma-define 98.1%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. associate--l- 98.7%

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

   6. sub-neg 98.7%

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

   7. neg-mul-1 98.7%

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

   8. distribute-lft-out 98.7%

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

   9. distribute-rgt-out 98.7%

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

   10. metadata-eval 98.7%

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

   11. distribute-lft-neg-out 98.7%

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

   12. distribute-rgt-neg-out 98.7%

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

   13. exp-prod 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in c around 0 98.1%

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

6. Final simplification 98.1%

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

Alternative 5: 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. Step-by-step derivation

   1. add-cube-cbrt 98.0%

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

   2. cbrt-unprod 98.0%

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

   3. frac-times 97.9%

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

   4. metadata-eval 97.9%

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

   5. add-sqr-sqrt 97.9%

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

   6. inv-pow 97.9%

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

   7. sqrt-pow2 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. metadata-eval 97.9%

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{\color{blue}{-0.5}}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right)} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{{\pi}^{-0.5}}\right) \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. cbrt-unprod 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\sqrt[3]{\frac{1}{\pi} \cdot {\pi}^{-0.5}}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. inv-pow 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\color{blue}{{\pi}^{-1}} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. metadata-eval 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{{\pi}^{\color{blue}{\left(-0.5 + -0.5\right)}} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. pow-prod-up 98.2%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \cdot {\pi}^{-0.5}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. add-cbrt-cube 98.2%

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

   7. associate--l- 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in c around 0 97.6%

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

   1. neg-mul-1 97.6%

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

   2. sub-neg 97.6%

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

   3. *-commutative 97.6%

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

   4. distribute-rgt-neg-in 97.6%

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

   5. metadata-eval 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

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

Alternative 6: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

   4. times-frac 98.7%

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

   5. *-lft-identity 98.7%

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

   6. associate--l- 98.7%

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

   7. sub-neg 98.7%

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

   8. neg-mul-1 98.7%

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

   9. distribute-lft-out 98.7%

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

   10. distribute-rgt-out 98.7%

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

   11. metadata-eval 98.7%

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

   12. exp-prod 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

   3. associate-*r* 96.1%

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

   4. *-commutative 96.1%

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

   5. associate-*l* 96.1%

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

   6. associate-+r+ 96.1%

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

   7. mul-1-neg 96.1%

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

   8. unsub-neg 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 7: 93.3% 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 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. Step-by-step derivation

   1. add-sqr-sqrt 97.4%

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

   2. sqrt-unprod 55.1%

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

   3. associate-*l/ 55.2%

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

   4. *-un-lft-identity 55.2%

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

   5. associate-*l/ 55.4%

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

   6. *-un-lft-identity 55.4%

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

   7. frac-times 55.0%

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

4. Applied egg-rr 49.9%

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

   1. associate-/l/ 50.2%

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

   2. +-commutative 50.2%

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

   3. +-commutative 50.2%

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

6. Simplified 50.2%

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

7. Taylor expanded in cosTheta around 0 93.4%

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

   1. mul-1-neg 93.4%

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

   2. associate-*r* 93.4%

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

   3. *-commutative 93.4%

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

   4. +-commutative 93.4%

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

9. Simplified 93.4%

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

10. Taylor expanded in c around inf 94.1%

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

    1. unsub-neg 94.1%

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

    2. *-commutative 94.1%

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

12. Applied egg-rr 94.1%

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

13. Final simplification 94.1%

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

Alternative 8: 93.3% 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. +-commutative 98.0%

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

   2. fma-define 98.1%

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

   3. associate-*l/ 98.7%

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

   4. *-lft-identity 98.7%

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

   5. associate--l- 98.7%

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

   6. sub-neg 98.7%

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

   7. neg-mul-1 98.7%

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

   8. distribute-lft-out 98.7%

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

   9. distribute-rgt-out 98.7%

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

   10. metadata-eval 98.7%

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

   11. distribute-lft-neg-out 98.7%

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

   12. distribute-rgt-neg-out 98.7%

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

   13. exp-prod 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in c around 0 98.1%

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

6. Taylor expanded in cosTheta around 0 94.1%

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

7. Final simplification 94.1%

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

Alternative 9: 10.9% accurate, 40.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = -cosTheta * (c * (cosTheta * 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. Step-by-step derivation

   1. add-sqr-sqrt 97.4%

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

   2. sqrt-unprod 55.1%

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

   3. associate-*l/ 55.2%

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

   4. *-un-lft-identity 55.2%

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

   5. associate-*l/ 55.4%

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

   6. *-un-lft-identity 55.4%

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

   7. frac-times 55.0%

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

4. Applied egg-rr 49.9%

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

   1. associate-/l/ 50.2%

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

   2. +-commutative 50.2%

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

   3. +-commutative 50.2%

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

6. Simplified 50.2%

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

7. Taylor expanded in cosTheta around 0 93.4%

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

   1. mul-1-neg 93.4%

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

   2. associate-*r* 93.4%

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

   3. *-commutative 93.4%

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

   4. +-commutative 93.4%

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

9. Simplified 93.4%

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

10. Taylor expanded in c around inf 94.1%

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

11. Taylor expanded in cosTheta around inf 11.1%

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

    1. mul-1-neg 11.1%

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

    2. *-commutative 11.1%

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

    3. *-commutative 11.1%

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

    4. distribute-lft-neg-out 11.1%

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

    5. *-commutative 11.1%

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

    6. distribute-rgt-neg-in 11.1%

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

13. Simplified 11.1%

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

14. Final simplification 11.1%

    \[\leadsto \left(-cosTheta\right) \cdot \left(c \cdot \left(cosTheta \cdot \pi\right)\right) \]
15. Add Preprocessing

Alternative 10: 5.0% accurate, 107.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. +-commutative 98.0%

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

   3. fma-define 98.1%

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

   4. times-frac 98.7%

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

   5. *-lft-identity 98.7%

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

   6. associate--l- 98.7%

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

   7. sub-neg 98.7%

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

   8. neg-mul-1 98.7%

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

   9. distribute-lft-out 98.7%

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

   10. distribute-rgt-out 98.7%

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

   11. metadata-eval 98.7%

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

   12. exp-prod 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in c around inf 4.9%

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

6. Final simplification 4.9%

   \[\leadsto \frac{1}{c} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024069 
(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))))))