Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 13.0s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / Float32(cosTheta * sqrt(Float32(pi)))), (exp(Float32(-cosTheta)) ^ cosTheta), 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-def 97.8%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (* (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 + (c + ((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(c + 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) + (c + ((exp(-(cosTheta ^ single(2.0))) * sqrt(((single(1.0) + (cosTheta * single(-2.0))) / single(pi)))) / cosTheta)));
end

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. fma-def 97.8%

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

   3. associate-*l/ 98.3%

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

   4. *-commutative 98.3%

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

   5. *-rgt-identity 98.3%

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

   6. associate--l- 98.3%

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

   7. count-2 98.3%

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

   8. *-commutative 98.3%

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

   9. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in c around 0 97.9%

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

   1. associate-*l/ 97.9%

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

   2. mul-1-neg 97.9%

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

   3. cancel-sign-sub-inv 97.9%

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

   4. metadata-eval 97.9%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (/
    (/ (sqrt (+ 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((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(sqrt(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0)))) / Float32(cosTheta * sqrt(Float32(pi)))) / exp(Float32(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 97.8%

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

   1. distribute-lft-neg-out 97.8%

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

   2. exp-neg 97.7%

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

   3. associate-*r/ 97.7%

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

   4. associate-/l* 97.7%

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

   5. /-rgt-identity 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. fma-def 97.8%

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

   3. associate-*l/ 98.3%

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

   4. *-commutative 98.3%

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

   5. *-rgt-identity 98.3%

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

   6. associate--l- 98.3%

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

   7. count-2 98.3%

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

   8. *-commutative 98.3%

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

   9. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in c around 0 97.9%

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

5. Taylor expanded in c around 0 97.5%

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

   1. *-commutative 97.5%

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

   2. cancel-sign-sub-inv 97.5%

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

   3. metadata-eval 97.5%

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

   4. *-commutative 97.5%

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

   5. neg-mul-1 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 5: 95.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. add-log-exp 97.0%

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

   4. associate--l- 97.0%

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

   5. exp-diff 96.9%

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

   6. prod-exp 96.9%

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

   7. exp-lft-sqr 96.9%

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

   8. *-commutative 96.9%

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

   9. exp-diff 97.0%

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

   10. add-log-exp 98.3%

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

   11. associate-/l/ 98.3%

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

   12. cancel-sign-sub-inv 98.3%

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

   13. metadata-eval 98.3%

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

   14. *-commutative 98.3%

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

   15. +-commutative 98.3%

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

   16. fma-udef 98.3%

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

3. Applied egg-rr 98.3%

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

4. Taylor expanded in cosTheta around 0 96.0%

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

   1. neg-mul-1 96.0%

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

   2. sub-neg 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 6: 94.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. fma-def 97.8%

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

   3. associate-*l/ 98.3%

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

   4. *-commutative 98.3%

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

   5. *-rgt-identity 98.3%

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

   6. associate--l- 98.3%

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

   7. count-2 98.3%

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

   8. *-commutative 98.3%

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

   9. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around 0 95.0%

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

   1. expm1-log1p-u 95.0%

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

   2. expm1-udef 21.6%

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

6. Applied egg-rr 21.6%

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

   1. expm1-def 95.0%

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

   2. expm1-log1p 95.0%

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 7: 94.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. +-commutative 97.8%

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

   2. fma-def 97.8%

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

   3. associate-*l/ 98.3%

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

   4. *-commutative 98.3%

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

   5. *-rgt-identity 98.3%

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

   6. associate--l- 98.3%

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

   7. count-2 98.3%

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

   8. *-commutative 98.3%

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

   9. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around 0 95.0%

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

   1. expm1-log1p-u 87.6%

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

   2. expm1-udef 87.6%

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

   3. distribute-rgt-out 87.6%

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

   4. pow1/2 87.6%

      \[\leadsto \frac{1}{1 + \left(c + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} - 1\right)\right)} \]

   5. inv-pow 87.6%

      \[\leadsto \frac{1}{1 + \left(c + \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} - 1\right)\right)} \]

   6. pow-pow 87.6%

      \[\leadsto \frac{1}{1 + \left(c + \left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} - 1\right)\right)} \]

   7. metadata-eval 87.6%

      \[\leadsto \frac{1}{1 + \left(c + \left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(-1 + \frac{1}{cosTheta}\right)\right)} - 1\right)\right)} \]

6. Applied egg-rr 87.6%

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

   1. expm1-def 87.6%

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

   2. expm1-log1p 95.0%

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 8: 94.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(-1.0) + (single(1.0) / cosTheta)) * 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. Step-by-step derivation

   1. +-commutative 97.8%

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

   2. fma-def 97.8%

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

   3. associate-*l/ 98.3%

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

   4. *-commutative 98.3%

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

   5. *-rgt-identity 98.3%

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

   6. associate--l- 98.3%

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

   7. count-2 98.3%

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

   8. *-commutative 98.3%

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

   9. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around 0 95.0%

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

5. Taylor expanded in c around 0 95.0%

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

   1. distribute-rgt-out 95.0%

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

7. Simplified 95.0%

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

8. Final simplification 95.0%

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

Alternative 9: 92.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. frac-times 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. add-log-exp 97.0%

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

   4. associate--l- 97.0%

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

   5. exp-diff 96.9%

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

   6. prod-exp 96.9%

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

   7. exp-lft-sqr 96.9%

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

   8. *-commutative 96.9%

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

   9. exp-diff 97.0%

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

   10. add-log-exp 98.3%

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

   11. associate-/l/ 98.3%

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

   12. cancel-sign-sub-inv 98.3%

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

   13. metadata-eval 98.3%

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

   14. *-commutative 98.3%

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

   15. +-commutative 98.3%

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

   16. fma-udef 98.3%

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

3. Applied egg-rr 98.3%

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

4. Taylor expanded in cosTheta around 0 92.2%

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

5. Final simplification 92.2%

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

Alternative 10: 10.8% accurate, 421.0× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. 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-def 97.8%

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

3. Simplified 98.3%

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

4. Taylor expanded in cosTheta around inf 11.0%

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

5. Taylor expanded in c around 0 11.1%

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

6. Final simplification 11.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023315 
(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))))))