Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 11.3s

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

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(sqrt(Float32(Float32(1.0) - Float32(cosTheta + cosTheta))) / Float32(cosTheta * sqrt(Float32(pi)))) * exp(Float32(cosTheta * Float32(-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 97.8%

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

   1. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--r+ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{\mathsf{fma}\left(cosTheta, -1.5, \mathsf{fma}\left(cosTheta, cosTheta \cdot 0.5, -1\right)\right) + \frac{1}{cosTheta}}{\sqrt{\pi}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (+
      (fma cosTheta -1.5 (fma cosTheta (* cosTheta 0.5) -1.0))
      (/ 1.0 cosTheta))
     (sqrt PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((fmaf(cosTheta, -1.5f, fmaf(cosTheta, (cosTheta * 0.5f), -1.0f)) + (1.0f / cosTheta)) / sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(fma(cosTheta, Float32(-1.5), fma(cosTheta, Float32(cosTheta * Float32(0.5)), Float32(-1.0))) + Float32(Float32(1.0) / cosTheta)) / sqrt(Float32(pi))))))
end

Derivation

1. Initial program 97.8%

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

   1. associate-+l+ 97.8%

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

   2. associate-*l* 97.8%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1}}{\sqrt{\pi}}\right)} \]
5. Step-by-step derivation

   1. pow1 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{\left(0.5 \cdot {cosTheta}^{2}\right)}^{1}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   2. unpow2 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left({\left(0.5 \cdot \color{blue}{\left(cosTheta \cdot cosTheta\right)}\right)}^{1} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{\left(0.5 \cdot \left(cosTheta \cdot cosTheta\right)\right)}^{1}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]
7. Step-by-step derivation

   1. unpow1 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{0.5 \cdot \left(cosTheta \cdot cosTheta\right)} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   2. unpow2 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot \color{blue}{{cosTheta}^{2}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   3. *-commutative 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{cosTheta}^{2} \cdot 0.5} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   4. unpow2 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{\left(cosTheta \cdot cosTheta\right)} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

8. Simplified 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{\left(cosTheta \cdot cosTheta\right) \cdot 0.5} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

9. Taylor expanded in cosTheta around 0 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1}}{\sqrt{\pi}}\right)} \]
10. Step-by-step derivation

    1. sub-neg 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) + \left(-1\right)}}{\sqrt{\pi}}\right)} \]

    2. fma-def 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\mathsf{fma}\left(-1.5, cosTheta, 0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)} + \left(-1\right)}{\sqrt{\pi}}\right)} \]

    3. *-commutative 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\mathsf{fma}\left(-1.5, cosTheta, \color{blue}{{cosTheta}^{2} \cdot 0.5} + \frac{1}{cosTheta}\right) + \left(-1\right)}{\sqrt{\pi}}\right)} \]

    4. unpow2 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\mathsf{fma}\left(-1.5, cosTheta, \color{blue}{\left(cosTheta \cdot cosTheta\right)} \cdot 0.5 + \frac{1}{cosTheta}\right) + \left(-1\right)}{\sqrt{\pi}}\right)} \]

    5. fma-udef 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\mathsf{fma}\left(-1.5, cosTheta, \color{blue}{\mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, \frac{1}{cosTheta}\right)}\right) + \left(-1\right)}{\sqrt{\pi}}\right)} \]

    6. metadata-eval 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\mathsf{fma}\left(-1.5, cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, \frac{1}{cosTheta}\right)\right) + \color{blue}{-1}}{\sqrt{\pi}}\right)} \]

    7. +-commutative 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{-1 + \mathsf{fma}\left(-1.5, cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, \frac{1}{cosTheta}\right)\right)}}{\sqrt{\pi}}\right)} \]

    8. fma-udef 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \color{blue}{\left(-1.5 \cdot cosTheta + \mathsf{fma}\left(cosTheta \cdot cosTheta, 0.5, \frac{1}{cosTheta}\right)\right)}}{\sqrt{\pi}}\right)} \]

    9. fma-udef 97.5%

       \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \left(-1.5 \cdot cosTheta + \color{blue}{\left(\left(cosTheta \cdot cosTheta\right) \cdot 0.5 + \frac{1}{cosTheta}\right)}\right)}{\sqrt{\pi}}\right)} \]

    10. unpow2 97.5%

        \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \left(-1.5 \cdot cosTheta + \left(\color{blue}{{cosTheta}^{2}} \cdot 0.5 + \frac{1}{cosTheta}\right)\right)}{\sqrt{\pi}}\right)} \]

    11. *-commutative 97.5%

        \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \left(-1.5 \cdot cosTheta + \left(\color{blue}{0.5 \cdot {cosTheta}^{2}} + \frac{1}{cosTheta}\right)\right)}{\sqrt{\pi}}\right)} \]

    12. associate-+r+ 97.5%

        \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \color{blue}{\left(\left(-1.5 \cdot cosTheta + 0.5 \cdot {cosTheta}^{2}\right) + \frac{1}{cosTheta}\right)}}{\sqrt{\pi}}\right)} \]

    13. associate-+r+ 97.5%

        \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1 + \left(-1.5 \cdot cosTheta + 0.5 \cdot {cosTheta}^{2}\right)\right) + \frac{1}{cosTheta}}}{\sqrt{\pi}}\right)} \]

11. Simplified 97.5%

    \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\mathsf{fma}\left(cosTheta, -1.5, \mathsf{fma}\left(cosTheta, cosTheta \cdot 0.5, -1\right)\right) + \frac{1}{cosTheta}}}{\sqrt{\pi}}\right)} \]

12. Final simplification 97.5%

    \[\leadsto \frac{1}{1 + \left(c + \frac{\mathsf{fma}\left(cosTheta, -1.5, \mathsf{fma}\left(cosTheta, cosTheta \cdot 0.5, -1\right)\right) + \frac{1}{cosTheta}}{\sqrt{\pi}}\right)} \]

Alternative 3: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{-1 + \left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + 0.5 \cdot \left(cosTheta \cdot cosTheta\right)\right)\right)}{\sqrt{\pi}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (+
      -1.0
      (+ (* cosTheta -1.5) (+ (/ 1.0 cosTheta) (* 0.5 (* cosTheta cosTheta)))))
     (sqrt PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((-1.0f + ((cosTheta * -1.5f) + ((1.0f / cosTheta) + (0.5f * (cosTheta * cosTheta))))) / sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(-1.0) + Float32(Float32(cosTheta * Float32(-1.5)) + Float32(Float32(Float32(1.0) / cosTheta) + Float32(Float32(0.5) * Float32(cosTheta * cosTheta))))) / sqrt(Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((single(-1.0) + ((cosTheta * single(-1.5)) + ((single(1.0) / cosTheta) + (single(0.5) * (cosTheta * cosTheta))))) / sqrt(single(pi)))));
end

Derivation

1. Initial program 97.8%

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

   1. associate-+l+ 97.8%

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

   2. associate-*l* 97.8%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) - 1}}{\sqrt{\pi}}\right)} \]
5. Step-by-step derivation

   1. pow1 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{\left(0.5 \cdot {cosTheta}^{2}\right)}^{1}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   2. unpow2 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left({\left(0.5 \cdot \color{blue}{\left(cosTheta \cdot cosTheta\right)}\right)}^{1} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{\left(0.5 \cdot \left(cosTheta \cdot cosTheta\right)\right)}^{1}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]
7. Step-by-step derivation

   1. unpow1 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{0.5 \cdot \left(cosTheta \cdot cosTheta\right)} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   2. unpow2 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(0.5 \cdot \color{blue}{{cosTheta}^{2}} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   3. *-commutative 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{{cosTheta}^{2} \cdot 0.5} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

   4. unpow2 97.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{\left(cosTheta \cdot cosTheta\right)} \cdot 0.5 + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

8. Simplified 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\left(-1.5 \cdot cosTheta + \left(\color{blue}{\left(cosTheta \cdot cosTheta\right) \cdot 0.5} + \frac{1}{cosTheta}\right)\right) - 1}{\sqrt{\pi}}\right)} \]

9. Final simplification 97.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + 0.5 \cdot \left(cosTheta \cdot cosTheta\right)\right)\right)}{\sqrt{\pi}}\right)} \]

Alternative 4: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)}{\sqrt{\pi}}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+ 1.0 (+ c (/ (+ -1.0 (+ (/ 1.0 cosTheta) (* cosTheta -1.5))) (sqrt PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((-1.0f + ((1.0f / cosTheta) + (cosTheta * -1.5f))) / sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) / cosTheta) + Float32(cosTheta * Float32(-1.5)))) / sqrt(Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((single(-1.0) + ((single(1.0) / cosTheta) + (cosTheta * single(-1.5)))) / sqrt(single(pi)))));
end

Derivation

1. Initial program 97.8%

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

   1. associate-+l+ 97.8%

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

   2. associate-*l* 97.8%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\left(-1.5 \cdot cosTheta + \frac{1}{cosTheta}\right) - 1}}{\sqrt{\pi}}\right)} \]

5. Final simplification 97.1%

   \[\leadsto \frac{1}{1 + \left(c + \frac{-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)}{\sqrt{\pi}}\right)} \]

Alternative 5: 95.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--r+ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in c around 0 97.1%

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

   1. fma-def 97.2%

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

   2. mul-1-neg 97.2%

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

   3. unpow2 97.2%

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

   4. cancel-sign-sub-inv 97.2%

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

   5. metadata-eval 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in cosTheta around 0 95.2%

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

   1. distribute-rgt-out 95.2%

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

9. Simplified 95.2%

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

10. Final simplification 95.2%

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

Alternative 6: 95.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-+l+ 97.8%

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

   2. associate-*l* 97.8%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 95.9%

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

5. Final simplification 95.9%

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

Alternative 7: 93.0% accurate, 2.1× 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 97.8%

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

   1. frac-times 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. associate--r+ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in cosTheta around 0 93.8%

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

5. Final simplification 93.8%

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

Alternative 8: 10.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. distribute-lft-neg-out 97.8%

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

   4. exp-neg 97.7%

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

   5. associate-*r/ 97.7%

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

   6. associate-/l* 97.7%

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

   7. /-rgt-identity 97.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in cosTheta around 0 96.4%

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

   1. fma-def 96.4%

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

   2. associate-*r* 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unpow2 96.4%

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

   5. distribute-rgt-neg-out 96.4%

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

   6. *-commutative 96.4%

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

   7. mul-1-neg 96.4%

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

6. Simplified 96.4%

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

7. Taylor expanded in c around inf 10.9%

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

   1. mul-1-neg 10.9%

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

   2. distribute-rgt-neg-in 10.9%

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

   3. distribute-lft-neg-out 10.9%

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

   4. unpow2 10.9%

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

   5. distribute-rgt-neg-out 10.9%

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

   6. associate-*l* 10.9%

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

9. Simplified 10.9%

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

10. Final simplification 10.9%

    \[\leadsto c \cdot \left(cosTheta \cdot \left(cosTheta \cdot \left(-\pi\right)\right)\right) \]

Alternative 9: 10.8% accurate, 140.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. distribute-lft-neg-out 97.8%

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

   4. exp-neg 97.7%

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

   5. associate-*r/ 97.7%

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

   6. associate-/l* 97.7%

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

   7. /-rgt-identity 97.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in cosTheta around inf 10.6%

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

5. Taylor expanded in c around 0 10.6%

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

   1. +-commutative 10.6%

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

   2. mul-1-neg 10.6%

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

   3. unsub-neg 10.6%

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

7. Simplified 10.6%

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

8. Final simplification 10.6%

   \[\leadsto 1 - c \]

Alternative 10: 10.8% accurate, 421.0× 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 97.8%

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

   1. +-commutative 97.8%

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

   2. associate-+l+ 97.8%

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

   3. distribute-lft-neg-out 97.8%

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

   4. exp-neg 97.7%

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

   5. associate-*r/ 97.7%

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

   6. associate-/l* 97.7%

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

   7. /-rgt-identity 97.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in cosTheta around inf 10.6%

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

5. Taylor expanded in c around 0 10.6%

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

6. Final simplification 10.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023172 
(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))))))