Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 11.6s

Alternatives: 14

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / Float32(sqrt(Float32(pi)) * Float32(cosTheta * exp(Float32(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. 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/r/ 98.1%

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

   6. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

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. Step-by-step derivation

   1. frac-times 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. associate--l- 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) * Float32(exp(Float32(cosTheta * Float32(-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. +-commutative 97.6%

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

   2. fma-def 97.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in c around 0 97.7%

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

   1. associate-+r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. +-commutative 97.7%

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

   4. *-commutative 97.7%

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

   5. fma-udef 97.7%

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

   6. mul-1-neg 97.7%

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

   7. unpow2 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) * exp(Float32(cosTheta * Float32(-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. +-commutative 97.6%

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

   2. fma-def 97.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in c around 0 97.7%

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

   1. associate-+r+ 97.7%

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

   2. *-commutative 97.7%

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

   3. +-commutative 97.7%

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

   4. *-commutative 97.7%

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

   5. fma-udef 97.7%

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

   6. mul-1-neg 97.7%

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

   7. unpow2 97.7%

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

6. Simplified 97.7%

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

7. Taylor expanded in c around 0 96.9%

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

   1. associate-*l/ 97.0%

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

   2. unpow2 97.0%

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

   3. +-commutative 97.0%

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

   4. *-commutative 97.0%

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

   5. fma-udef 97.0%

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

9. Simplified 97.0%

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

10. Final simplification 97.0%

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

Alternative 5: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{\left(cosTheta \cdot -0.5 + \left(-0.5 \cdot {cosTheta}^{2} + \frac{1}{cosTheta}\right)\right) + -1}{\sqrt{\pi} \cdot \left(1 + cosTheta \cdot cosTheta\right)}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (+
      (+ (* cosTheta -0.5) (+ (* -0.5 (pow cosTheta 2.0)) (/ 1.0 cosTheta)))
      -1.0)
     (* (sqrt PI) (+ 1.0 (* cosTheta cosTheta))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((((cosTheta * -0.5f) + ((-0.5f * powf(cosTheta, 2.0f)) + (1.0f / cosTheta))) + -1.0f) / (sqrtf(((float) M_PI)) * (1.0f + (cosTheta * cosTheta))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(Float32(cosTheta * Float32(-0.5)) + Float32(Float32(Float32(-0.5) * (cosTheta ^ Float32(2.0))) + Float32(Float32(1.0) / cosTheta))) + Float32(-1.0)) / Float32(sqrt(Float32(pi)) * Float32(Float32(1.0) + Float32(cosTheta * cosTheta)))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((((cosTheta * single(-0.5)) + ((single(-0.5) * (cosTheta ^ single(2.0))) + (single(1.0) / cosTheta))) + single(-1.0)) / (sqrt(single(pi)) * (single(1.0) + (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. 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/l* 98.1%

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

   6. /-rgt-identity 98.1%

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

   7. associate-/l* 98.1%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.2%

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

   1. distribute-rgt1-in 97.2%

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

   2. unpow2 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in cosTheta around 0 96.9%

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

8. Final simplification 96.9%

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

Alternative 6: 96.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((((cosTheta * single(-0.5)) + (single(1.0) / cosTheta)) + single(-1.0)) / (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. 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/l* 98.1%

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

   6. /-rgt-identity 98.1%

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

   7. associate-/l* 98.1%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 96.4%

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

5. Final simplification 96.4%

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

Alternative 7: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + ((((cosTheta * single(-0.5)) + (single(1.0) / cosTheta)) + single(-1.0)) / (sqrt(single(pi)) * (single(1.0) + (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. 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/l* 98.1%

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

   6. /-rgt-identity 98.1%

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

   7. associate-/l* 98.1%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.2%

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

   1. distribute-rgt1-in 97.2%

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

   2. unpow2 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in cosTheta around 0 96.4%

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

8. Final simplification 96.4%

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

Alternative 8: 96.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((1.0f / (cosTheta + (cosTheta * (cosTheta * 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(Float32(1.0) / Float32(cosTheta + Float32(cosTheta * Float32(cosTheta * 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) + ((single(1.0) / (cosTheta + (cosTheta * (cosTheta * 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/l* 98.1%

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

   6. /-rgt-identity 98.1%

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

   7. associate-/l* 98.1%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.2%

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

   1. distribute-rgt1-in 97.2%

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

   2. unpow2 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in c around 0 96.1%

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

   1. distribute-rgt-in 96.1%

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

   2. *-un-lft-identity 96.1%

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

   3. unpow2 96.1%

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

9. Applied egg-rr 96.1%

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

10. Final simplification 96.1%

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

Alternative 9: 95.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (((single(1.0) / cosTheta) + single(-1.0)) / (sqrt(single(pi)) * (single(1.0) + (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. 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/l* 98.1%

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

   6. /-rgt-identity 98.1%

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

   7. associate-/l* 98.1%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.2%

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

   1. distribute-rgt1-in 97.2%

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

   2. unpow2 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in cosTheta around 0 95.2%

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

8. Final simplification 95.2%

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

Alternative 10: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (((single(1.0) / cosTheta) + single(-1.0)) / 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/l* 98.1%

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

   6. /-rgt-identity 98.1%

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

   7. associate-/l* 98.1%

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

3. Simplified 98.2%

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

4. Taylor expanded in cosTheta around 0 97.2%

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

   1. distribute-rgt1-in 97.2%

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

   2. unpow2 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in cosTheta around 0 95.2%

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

8. Taylor expanded in cosTheta around 0 94.8%

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

9. Final simplification 94.8%

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

Alternative 11: 92.8% 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 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/r/ 98.1%

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

   6. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 92.5%

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

5. Final simplification 92.5%

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

Alternative 12: 10.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\left(c \cdot c - {c}^{3}\right) - c\right) \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (+ 1.0 (- (- (* c c) (pow c 3.0)) c)))

C

float code(float cosTheta, float c) {
	return 1.0f + (((c * c) - powf(c, 3.0f)) - c);
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 + (((c * c) - (c ** 3.0e0)) - c)
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) + Float32(Float32(Float32(c * c) - (c ^ Float32(3.0))) - c))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) + (((c * c) - (c ^ single(3.0))) - 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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/r/ 98.1%

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

   6. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around inf 10.6%

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

5. Taylor expanded in c around 0 10.6%

   \[\leadsto \color{blue}{1 + \left(-1 \cdot c + \left(-1 \cdot {c}^{3} + {c}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 10.6%

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

   2. mul-1-neg 10.6%

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

   3. unsub-neg 10.6%

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

   4. +-commutative 10.6%

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

   5. mul-1-neg 10.6%

      \[\leadsto 1 + \left(\left({c}^{2} + \color{blue}{\left(-{c}^{3}\right)}\right) - c\right) \]

   6. unsub-neg 10.6%

      \[\leadsto 1 + \left(\color{blue}{\left({c}^{2} - {c}^{3}\right)} - c\right) \]

   7. unpow2 10.6%

      \[\leadsto 1 + \left(\left(\color{blue}{c \cdot c} - {c}^{3}\right) - c\right) \]

7. Simplified 10.6%

   \[\leadsto \color{blue}{1 + \left(\left(c \cdot c - {c}^{3}\right) - c\right)} \]

8. Final simplification 10.6%

   \[\leadsto 1 + \left(\left(c \cdot c - {c}^{3}\right) - c\right) \]

Alternative 13: 10.9% accurate, 140.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/r/ 98.1%

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

   6. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around inf 10.6%

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

5. Taylor expanded in c around 0 10.6%

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

   1. mul-1-neg 10.6%

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

   2. unsub-neg 10.6%

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

7. Simplified 10.6%

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

8. Final simplification 10.6%

   \[\leadsto 1 - c \]

Alternative 14: 10.9% 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 97.6%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. associate-*l* 97.7%

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

   3. associate-*l/ 98.2%

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

   4. *-lft-identity 98.2%

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

   5. associate-/r/ 98.1%

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

   6. associate-/l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around inf 10.6%

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

5. Taylor expanded in c around 0 10.6%

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

6. Final simplification 10.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(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))))))