Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 18.6s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((sqrtf((1.0f + (cosTheta * -2.0f))) / (cosTheta * sqrtf(((float) M_PI)))) / powf(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 * Float32(-2.0)))) / Float32(cosTheta * sqrt(Float32(pi)))) / (exp(cosTheta) ^ cosTheta))))
end

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. distribute-lft-neg-out 97.6%

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

   2. exp-neg 97.6%

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

   3. associate-*r/ 97.6%

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

   4. associate-/l* 97.6%

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

   5. /-rgt-identity 97.6%

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

   6. *-commutative 97.6%

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

   7. *-commutative 97.6%

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

   8. times-frac 98.6%

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

   9. *-lft-identity 98.6%

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

3. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 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 97.6%

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

   1. 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--l- 98.6%

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

   4. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in c around 0 97.7%

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

6. Final simplification 97.7%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in c around 0 96.7%

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

6. Final simplification 96.7%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. 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--l- 98.6%

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

   4. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in c around 0 96.7%

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

   1. neg-mul-1 96.7%

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

   2. *-commutative 96.7%

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

7. Simplified 96.7%

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

   1. associate-*l/ 96.9%

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

   2. *-commutative 96.9%

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

   3. cancel-sign-sub-inv 96.9%

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

   4. metadata-eval 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 6: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \frac{1}{1 + \left(c + \left(cosTheta \cdot \left(t_0 \cdot -1.5\right) + t_0 \cdot \left(\frac{1}{cosTheta} + -1\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (/
    1.0
    (+
     1.0
     (+ c (+ (* cosTheta (* t_0 -1.5)) (* t_0 (+ (/ 1.0 cosTheta) -1.0))))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	return 1.0f / (1.0f + (c + ((cosTheta * (t_0 * -1.5f)) + (t_0 * ((1.0f / cosTheta) + -1.0f)))));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(cosTheta * Float32(t_0 * Float32(-1.5))) + Float32(t_0 * Float32(Float32(Float32(1.0) / cosTheta) + Float32(-1.0)))))))
end

MATLAB

function tmp = code(cosTheta, c)
	t_0 = sqrt((single(1.0) / single(pi)));
	tmp = single(1.0) / (single(1.0) + (c + ((cosTheta * (t_0 * single(-1.5))) + (t_0 * ((single(1.0) / cosTheta) + single(-1.0))))));
end

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in c around 0 97.7%

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

6. Taylor expanded in cosTheta around 0 96.2%

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

   1. +-commutative 96.2%

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

   2. associate-+r+ 96.2%

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

   3. +-commutative 96.2%

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

   4. distribute-rgt-out 96.2%

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

   5. distribute-rgt-out 96.2%

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

   6. metadata-eval 96.2%

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

8. Simplified 96.2%

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

9. Final simplification 96.2%

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

Alternative 7: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. 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--l- 98.6%

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

   4. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in c around 0 96.7%

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

   1. neg-mul-1 96.7%

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

   2. *-commutative 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in cosTheta around 0 95.6%

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

   1. +-commutative 95.6%

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

   2. +-commutative 95.6%

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

   3. associate-+l+ 95.6%

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

   4. associate-*l/ 96.0%

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

   5. *-lft-identity 96.0%

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

10. Simplified 96.0%

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

    1. clear-num 96.0%

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

    2. inv-pow 96.0%

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

    3. pow1/2 96.0%

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

    4. inv-pow 96.0%

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

    5. pow-pow 96.0%

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

    6. metadata-eval 96.0%

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

12. Applied egg-rr 96.0%

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

    1. unpow-1 96.0%

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

14. Simplified 96.0%

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

15. Final simplification 96.0%

    \[\leadsto \frac{1}{1 + \left(\frac{1}{\frac{cosTheta}{{\pi}^{-0.5}}} + \sqrt{\frac{1}{\pi}} \cdot \left(-1 + cosTheta \cdot -1.5\right)\right)} \]
16. Add Preprocessing

Alternative 8: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. 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--l- 98.6%

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

   4. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in c around 0 96.7%

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

   1. neg-mul-1 96.7%

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

   2. *-commutative 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in cosTheta around 0 95.6%

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

   1. +-commutative 95.6%

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

   2. +-commutative 95.6%

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

   3. associate-+l+ 95.6%

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

   4. associate-*l/ 96.0%

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

   5. *-lft-identity 96.0%

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

10. Simplified 96.0%

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

    1. *-un-lft-identity 96.0%

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

    2. pow1/2 96.0%

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

    3. inv-pow 96.0%

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

    4. pow-pow 96.0%

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

    5. metadata-eval 96.0%

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

12. Applied egg-rr 96.0%

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

    1. *-lft-identity 96.0%

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

14. Simplified 96.0%

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

15. Final simplification 96.0%

    \[\leadsto \frac{1}{1 + \left(\sqrt{\frac{1}{\pi}} \cdot \left(-1 + cosTheta \cdot -1.5\right) + \frac{{\pi}^{-0.5}}{cosTheta}\right)} \]
16. Add Preprocessing

Alternative 9: 95.9% accurate, 1.5× speedup

Math

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

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(cosTheta * Float32(-cosTheta))) * Float32(Float32(Float32(1.0) - cosTheta) / Float32(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 97.6%

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

   1. 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--l- 98.6%

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

   4. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 10: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. 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--l- 98.6%

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

   4. *-commutative 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in c around 0 96.7%

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

   1. neg-mul-1 96.7%

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

   2. *-commutative 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in cosTheta around 0 95.6%

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

   1. +-commutative 95.6%

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

   2. +-commutative 95.6%

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

   3. associate-+l+ 95.6%

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

   4. associate-*l/ 96.0%

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

   5. *-lft-identity 96.0%

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

10. Simplified 96.0%

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

    1. expm1-log1p-u 96.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \left(\frac{\sqrt{\frac{1}{\pi}}}{cosTheta} + \sqrt{\frac{1}{\pi}} \cdot \left(-1.5 \cdot cosTheta + -1\right)\right)}\right)\right)} \]

    2. expm1-udef 22.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \left(\frac{\sqrt{\frac{1}{\pi}}}{cosTheta} + \sqrt{\frac{1}{\pi}} \cdot \left(-1.5 \cdot cosTheta + -1\right)\right)}\right)} - 1} \]

12. Applied egg-rr 22.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5}, \mathsf{fma}\left(cosTheta, -1.5, -1\right), \frac{{\pi}^{-0.5}}{cosTheta}\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 96.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \mathsf{fma}\left({\pi}^{-0.5}, \mathsf{fma}\left(cosTheta, -1.5, -1\right), \frac{{\pi}^{-0.5}}{cosTheta}\right)}\right)\right)} \]

    2. expm1-log1p 96.0%

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

    3. fma-udef 96.0%

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

    4. *-rgt-identity 96.0%

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

    5. associate-*r/ 95.6%

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

    6. distribute-lft-out 95.7%

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

14. Simplified 95.7%

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

15. Final simplification 95.7%

    \[\leadsto \frac{1}{1 + {\pi}^{-0.5} \cdot \left(\frac{1}{cosTheta} + \mathsf{fma}\left(cosTheta, -1.5, -1\right)\right)} \]
16. Add Preprocessing

Alternative 11: 95.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in cosTheta around 0 94.6%

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

   1. +-commutative 94.6%

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

   2. distribute-rgt-out 94.7%

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

   3. fma-def 94.7%

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

7. Simplified 94.7%

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

8. Final simplification 94.7%

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

Alternative 12: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in cosTheta around 0 94.6%

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

   1. distribute-rgt-out 94.7%

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

7. Simplified 94.7%

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

8. Final simplification 94.7%

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

Alternative 13: 92.9% 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 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in cosTheta around 0 92.2%

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

6. Final simplification 92.2%

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

Alternative 14: 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 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in cosTheta around inf 11.0%

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

6. Taylor expanded in c around 0 11.0%

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

   1. mul-1-neg 11.0%

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

   2. unsub-neg 11.0%

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

8. Simplified 11.0%

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

9. Final simplification 11.0%

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

Alternative 15: 10.8% accurate, 322.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. +-commutative 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 97.6%

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

   5. fma-def 97.6%

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

3. Simplified 97.7%

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

5. Taylor expanded in cosTheta around inf 11.0%

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

6. Taylor expanded in c around 0 11.0%

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

7. Final simplification 11.0%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))