Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.4%

Time: 18.9s

Alternatives: 11

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((sqrt((cosTheta * ((single(1.0) / cosTheta) + single(-2.0)))) / 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. Add Preprocessing

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. sub-neg 98.5%

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

   4. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

6. Final simplification 98.5%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp(-(cosTheta * cosTheta)) * (sqrt((single(1.0) + (cosTheta * single(-2.0)))) / (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. Add Preprocessing

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. frac-times 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. sub-neg 98.2%

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

   4. metadata-eval 98.2%

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

   5. *-commutative 98.2%

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

5. Applied egg-rr 98.2%

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

   1. distribute-lft-in 98.4%

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

   2. pow1 98.4%

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

   3. inv-pow 98.4%

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

   4. pow-prod-up 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{{cosTheta}^{\color{blue}{0}} + cosTheta \cdot -2}}{cosTheta}}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp(-(cosTheta * cosTheta)) * ((sqrt((single(1.0) + (cosTheta * single(-2.0)))) / 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. Add Preprocessing

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. sub-neg 98.5%

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

   4. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. distribute-lft-in 98.4%

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

   2. pow1 98.4%

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

   3. inv-pow 98.4%

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

   4. pow-prod-up 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{{cosTheta}^{\color{blue}{0}} + cosTheta \cdot -2}}{cosTheta}}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 4: 97.9% 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 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. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around 0 98.0%

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

6. Final simplification 98.0%

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

Alternative 5: 97.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp(-(cosTheta * cosTheta)) * (((single(1.0) + (cosTheta * (single(-1.0) + (cosTheta * ((cosTheta * single(-0.5)) - single(0.5)))))) / 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. Add Preprocessing

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. sub-neg 98.5%

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

   4. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. distribute-lft-in 98.4%

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

   2. pow1 98.4%

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

   3. inv-pow 98.4%

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

   4. pow-prod-up 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{{cosTheta}^{\color{blue}{0}} + cosTheta \cdot -2}}{cosTheta}}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in cosTheta around 0 97.5%

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

9. Final simplification 97.5%

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

Alternative 6: 96.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp(-(cosTheta * cosTheta)) * (((single(1.0) + (cosTheta * (single(-1.0) + (cosTheta * single(-0.5))))) / 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. Add Preprocessing

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. sub-neg 98.5%

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

   4. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. distribute-lft-in 98.4%

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

   2. pow1 98.4%

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

   3. inv-pow 98.4%

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

   4. pow-prod-up 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{{cosTheta}^{\color{blue}{0}} + cosTheta \cdot -2}}{cosTheta}}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in cosTheta around 0 97.3%

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

9. Final simplification 97.3%

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

Alternative 7: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (exp(-(cosTheta * cosTheta)) * (((single(1.0) - 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. Add Preprocessing

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. sub-neg 98.5%

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

   4. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

   1. distribute-lft-in 98.4%

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

   2. pow1 98.4%

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

   3. inv-pow 98.4%

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

   4. pow-prod-up 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{{cosTheta}^{\color{blue}{0}} + cosTheta \cdot -2}}{cosTheta}}{\sqrt{\pi}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in cosTheta around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

10. Simplified 96.4%

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

11. Final simplification 96.4%

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

Alternative 8: 95.8% 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 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. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in cosTheta around 0 96.1%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right)} \]
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 9: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in cosTheta around inf 97.8%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

   3. sub-neg 98.5%

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

   4. metadata-eval 98.5%

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

5. Applied egg-rr 98.5%

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

6. Taylor expanded in c around 0 55.9%

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

   1. sub-neg 55.9%

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

   2. metadata-eval 55.9%

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

   3. neg-mul-1 55.9%

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

8. Simplified 55.9%

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

9. Taylor expanded in cosTheta around 0 95.7%

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

    1. mul-1-neg 95.7%

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

    2. unsub-neg 95.7%

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

    3. associate-*r* 95.7%

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

    4. mul-1-neg 95.7%

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

    5. sub-neg 95.7%

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

11. Simplified 95.7%

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

12. Final simplification 95.7%

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

Alternative 10: 93.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \sqrt{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. add-cube-cbrt 97.8%

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

   2. cbrt-unprod 97.8%

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

   3. frac-times 98.2%

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

   4. metadata-eval 98.2%

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

   5. add-sqr-sqrt 98.2%

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

   6. inv-pow 98.2%

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

   7. sqrt-pow2 98.2%

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

   8. metadata-eval 98.2%

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in cosTheta around 0 92.7%

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

6. Final simplification 92.7%

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

Alternative 11: 10.8% accurate, 107.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. +-commutative 97.8%

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

   3. fma-define 97.8%

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

   4. times-frac 98.4%

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

   5. *-lft-identity 98.4%

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

   6. associate--l- 98.4%

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

   7. sub-neg 98.4%

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

   8. neg-mul-1 98.4%

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

   9. distribute-lft-out 98.4%

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

   10. distribute-rgt-out 98.4%

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

   11. metadata-eval 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in c around inf 10.7%

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

6. Taylor expanded in c around 0 10.8%

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

   1. mul-1-neg 10.8%

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

   2. unsub-neg 10.8%

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

8. Simplified 10.8%

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

9. Final simplification 10.8%

   \[\leadsto 1 - c \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))))