Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 12.9s

Alternatives: 14

Speedup: 0.8×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{\sqrt[3]{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{1.5}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (cbrt (pow (fma cosTheta -2.0 1.0) 1.5))
     (* (sqrt PI) (* cosTheta (exp (* cosTheta cosTheta)))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (cbrtf(powf(fmaf(cosTheta, -2.0f, 1.0f), 1.5f)) / (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(cbrt((fma(cosTheta, Float32(-2.0), Float32(1.0)) ^ Float32(1.5))) / Float32(sqrt(Float32(pi)) * Float32(cosTheta * exp(Float32(cosTheta * cosTheta))))))))
end

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

   1. add-cbrt-cube 98.6%

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

   2. add-sqr-sqrt 98.5%

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

5. Applied egg-rr 98.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\sqrt[3]{\mathsf{fma}\left(cosTheta, -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.5%

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

   2. unpow1/2 98.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.5}} \cdot \mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

   3. pow-plus 98.6%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{\left(0.5 + 1\right)}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

   4. metadata-eval 98.6%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{\color{blue}{1.5}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

7. Simplified 98.6%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{1.5}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

8. Final simplification 98.6%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{1.5}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

Alternative 2: 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 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(Float32(1.0) / Float32(cosTheta * exp((cosTheta ^ Float32(2.0))))))))
end

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in c around 0 98.0%

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

5. Final simplification 98.0%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in c around 0 98.0%

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

3. Final simplification 98.0%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. add-sqr-sqrt 97.8%

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

5. Applied egg-rr 97.8%

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

6. Taylor expanded in c around 0 98.0%

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

   1. +-commutative 98.0%

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

   2. associate-*r/ 97.9%

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

   3. +-commutative 97.9%

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

   4. *-commutative 97.9%

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

   5. fma-udef 97.9%

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

   6. *-rgt-identity 97.9%

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

   7. *-commutative 97.9%

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

   8. unpow2 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 6: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(c + \frac{\left(1 - cosTheta\right) + \left(cosTheta \cdot cosTheta\right) \cdot -0.5}{\sqrt{\pi} \cdot \left(cosTheta + {cosTheta}^{3}\right)}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (+
    c
    (/
     (+ (- 1.0 cosTheta) (* (* cosTheta cosTheta) -0.5))
     (* (sqrt PI) (+ cosTheta (pow cosTheta 3.0))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + (((1.0f - cosTheta) + ((cosTheta * cosTheta) * -0.5f)) / (sqrtf(((float) M_PI)) * (cosTheta + powf(cosTheta, 3.0f))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(Float32(Float32(Float32(1.0) - cosTheta) + Float32(Float32(cosTheta * cosTheta) * Float32(-0.5))) / Float32(sqrt(Float32(pi)) * Float32(cosTheta + (cosTheta ^ Float32(3.0))))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + (c + (((single(1.0) - cosTheta) + ((cosTheta * cosTheta) * single(-0.5))) / (sqrt(single(pi)) * (cosTheta + (cosTheta ^ single(3.0)))))));
end

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 97.0%

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

   1. neg-mul-1 97.0%

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

   2. associate-+r+ 97.0%

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

   3. sub-neg 97.0%

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

   4. *-commutative 97.0%

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

   5. unpow2 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in cosTheta around 0 97.0%

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

8. Final simplification 97.0%

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

Alternative 7: 96.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 96.9%

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

5. Final simplification 96.9%

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

Alternative 8: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + (c + ((-1.0f + ((cosTheta * -0.5f) + (1.0f / cosTheta))) / (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(-1.0) + Float32(Float32(cosTheta * Float32(-0.5)) + Float32(Float32(1.0) / cosTheta))) / 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(-0.5)) + (single(1.0) / cosTheta))) / (sqrt(single(pi)) * (single(1.0) + (cosTheta * cosTheta))))));
end

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/l* 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta around 0 96.9%

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

5. Taylor expanded in cosTheta around 0 97.0%

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

   1. distribute-rgt1-in 96.9%

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

   2. unpow2 96.9%

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

7. Simplified 96.9%

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

8. Final simplification 96.9%

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

Alternative 9: 96.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around 0 97.0%

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

   1. neg-mul-1 97.0%

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

   2. associate-+r+ 97.0%

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

   3. sub-neg 97.0%

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

   4. *-commutative 97.0%

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

   5. unpow2 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in c around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. *-commutative 96.4%

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

   3. unpow2 96.4%

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

   4. associate-*r* 96.4%

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

   5. *-commutative 96.4%

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

   6. unpow2 96.4%

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

9. Simplified 96.4%

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

10. Taylor expanded in cosTheta around 0 96.4%

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

11. Final simplification 96.4%

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

Alternative 10: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.0%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.5%

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

   1. +-commutative 95.5%

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

   2. distribute-rgt-out 95.5%

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

   3. fma-def 95.5%

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

6. Simplified 95.5%

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

   1. fma-udef 95.5%

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

   2. sqrt-div 95.5%

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

   3. metadata-eval 95.5%

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

   4. inv-pow 95.5%

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

8. Applied egg-rr 95.5%

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

   1. +-commutative 95.5%

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

   2. associate-*l/ 95.9%

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

   3. *-lft-identity 95.9%

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

   4. unpow-1 95.9%

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

10. Simplified 95.9%

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

11. Final simplification 95.9%

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

Alternative 11: 95.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((-1.0f + (1.0f / cosTheta)) / sqrtf(((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(pi)))))
end

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.0%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.5%

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

   1. +-commutative 95.5%

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

   2. distribute-rgt-out 95.5%

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

   3. fma-def 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in c around 0 95.4%

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

   1. pow1 95.4%

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

   2. sub-neg 95.4%

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

   3. metadata-eval 95.4%

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

   4. sqrt-div 95.4%

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

   5. metadata-eval 95.4%

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

9. Applied egg-rr 95.4%

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

    1. unpow1 95.4%

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

    2. associate-*r/ 95.8%

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

    3. *-rgt-identity 95.8%

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

    4. +-commutative 95.8%

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

11. Simplified 95.8%

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

12. Final simplification 95.8%

    \[\leadsto \frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \]

Alternative 12: 93.1% 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 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

   1. add-cbrt-cube 98.6%

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

   2. add-sqr-sqrt 98.5%

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

5. Applied egg-rr 98.5%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\sqrt[3]{\mathsf{fma}\left(cosTheta, -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.5%

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

   2. unpow1/2 98.5%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.5}} \cdot \mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

   3. pow-plus 98.6%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{\left(0.5 + 1\right)}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

   4. metadata-eval 98.6%

      \[\leadsto \frac{1}{1 + \left(c + \frac{\sqrt[3]{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{\color{blue}{1.5}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

7. Simplified 98.6%

   \[\leadsto \frac{1}{1 + \left(c + \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{1.5}}}}{\sqrt{\pi} \cdot \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]

8. Taylor expanded in cosTheta around 0 93.2%

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

9. Final simplification 93.2%

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

Alternative 13: 10.8% accurate, 140.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta around inf 10.8%

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

5. Taylor expanded in c around 0 10.8%

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

   1. +-commutative 10.8%

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

   2. mul-1-neg 10.8%

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

   3. unsub-neg 10.8%

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

7. Simplified 10.8%

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

8. Final simplification 10.8%

   \[\leadsto 1 - c \]

Alternative 14: 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 98.0%

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

   1. associate-+l+ 98.0%

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

   2. associate-*l* 97.9%

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

   3. associate-*l/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-/r/ 98.5%

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

   6. associate-/l/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in c around 0 98.0%

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

5. Taylor expanded in cosTheta around inf 10.8%

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

6. Final simplification 10.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023309 
(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))))))