Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.6%

Time: 11.0s

Alternatives: 12

Speedup: 1.1×

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: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-lft-identity N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-lft-identity N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.6

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

4. Applied rewrites 98.6%

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

6. Applied rewrites 98.1%

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

Alternative 3: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. rec-exp N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

7. Applied rewrites 97.8%

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

Alternative 4: 97.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, cosTheta \cdot cosTheta, 0.5\right), cosTheta \cdot cosTheta, -1\right), cosTheta \cdot cosTheta, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{cosTheta}, 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (fma
    (fma
     (fma -0.16666666666666666 (* cosTheta cosTheta) 0.5)
     (* cosTheta cosTheta)
     -1.0)
    (* cosTheta cosTheta)
    1.0)
   (/ (sqrt (/ (fma -2.0 cosTheta 1.0) PI)) cosTheta)
   1.0)))

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf(fmaf(fmaf(fmaf(-0.16666666666666666f, (cosTheta * cosTheta), 0.5f), (cosTheta * cosTheta), -1.0f), (cosTheta * cosTheta), 1.0f), (sqrtf((fmaf(-2.0f, cosTheta, 1.0f) / ((float) M_PI))) / cosTheta), 1.0f);
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(fma(fma(fma(Float32(-0.16666666666666666), Float32(cosTheta * cosTheta), Float32(0.5)), Float32(cosTheta * cosTheta), Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(1.0)), Float32(sqrt(Float32(fma(Float32(-2.0), cosTheta, Float32(1.0)) / Float32(pi))) / cosTheta), Float32(1.0)))
end

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. rec-exp N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 97.7%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, cosTheta \cdot cosTheta, 0.5\right), cosTheta \cdot cosTheta, -1\right), cosTheta \cdot cosTheta, 1\right), \frac{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}}{cosTheta}, 1\right)} \]
11. Add Preprocessing

Alternative 5: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, -1\right), cosTheta \cdot cosTheta, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}{cosTheta}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. rec-exp N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 97.6%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, cosTheta \cdot cosTheta, -1\right), cosTheta \cdot cosTheta, 1\right), \frac{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}}}{cosTheta}, 1\right)} \]
11. Add Preprocessing

Alternative 6: 96.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. rec-exp N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 97.1%

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

Alternative 7: 97.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 97.1

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

7. Applied rewrites 97.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-1.5, cosTheta, -1\right), cosTheta, 1\right)}{cosTheta}}}}} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

9. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-1.5, cosTheta, -1\right), cosTheta, 1\right)}{cosTheta}}{\sqrt{\pi}} + \left(c + 1\right)}} \]
10. Add Preprocessing

Alternative 8: 95.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 96.1%

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

Alternative 9: 95.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. mul-1-neg N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\left(1 + \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}} \]

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 95.4

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

7. Applied rewrites 95.4%

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

9. Applied rewrites 95.9%

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

10. Final simplification 95.9%

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

Alternative 10: 95.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. exp-neg N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. mul-1-neg N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\left(1 + \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{cosTheta}} \]

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 95.4

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

7. Applied rewrites 95.4%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

9. Applied rewrites 95.9%

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

Alternative 11: 93.1% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot cosTheta \]

   4. lower-PI.f32 92.5

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

5. Applied rewrites 92.5%

   \[\leadsto \color{blue}{\sqrt{\pi} \cdot cosTheta} \]
6. Add Preprocessing

Alternative 12: 5.1% accurate, 15.3× speedup

Math

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

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 97.7%

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

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{1}{c}} \]
4. Step-by-step derivation

   1. lower-/.f32 5.0

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

5. Applied rewrites 5.0%

   \[\leadsto \color{blue}{\frac{1}{c}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))))