Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 4.0s

Alternatives: 15

Speedup: 1.2×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.4%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

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

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

   5. unpow2 N/A

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

   6. exp-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 97.8

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

3. Applied rewrites 97.8%

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

4. Taylor expanded in c around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. pow2 N/A

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

   3. rec-exp N/A

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

   4. pow2 N/A

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

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

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. lower-+.f32 N/A

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

   8. lower-+.f32 N/A

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

6. Applied rewrites 98.5%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 97.9%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. count-2-rev N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 97.9

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

9. Applied rewrites 97.9%

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

Alternative 4: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 97.9%

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

Alternative 5: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in c around 0

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

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

4. Applied rewrites 97.9%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

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

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

   5. unpow2 N/A

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

   6. exp-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 97.8

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

3. Applied rewrites 97.8%

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

4. Taylor expanded in c around 0

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

   1. fp-cancel-sign-sub-inv N/A

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

   2. pow2 N/A

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

   3. rec-exp N/A

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

   4. pow2 N/A

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

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

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. lower-+.f32 N/A

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

6. Applied rewrites 98.0%

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

Alternative 7: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

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

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

   5. lift-neg.f32 N/A

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

   6. lower-fma.f32 97.2

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

6. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-lft-identity N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

8. Applied rewrites 97.3%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. lower--.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. lower--.f32 N/A

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

    9. lower-*.f32 97.3

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

11. Applied rewrites 97.3%

    \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\mathsf{fma}\left(\left(0.5 \cdot cosTheta - 1.5\right) \cdot cosTheta - 1, cosTheta, 1\right)}{cosTheta}}}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 8: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

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

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

   5. lift-neg.f32 N/A

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

   6. lower-fma.f32 97.2

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

6. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-lft-identity N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

8. Applied rewrites 97.3%

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

Alternative 9: 97.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

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

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

   5. lift-neg.f32 N/A

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

   6. lower-fma.f32 97.2

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

6. Applied rewrites 97.2%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

8. Applied rewrites 97.4%

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

Alternative 10: 96.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

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

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-fma.f32 96.8

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

10. Applied rewrites 96.8%

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

Alternative 11: 96.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

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

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

   5. lift-neg.f32 N/A

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

   6. lower-fma.f32 97.2

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

6. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-lft-identity N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

8. Applied rewrites 97.3%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. lower--.f32 N/A

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

    6. lower-*.f32 96.7

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

11. Applied rewrites 96.7%

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

Alternative 12: 95.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 95.5%

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

Alternative 13: 95.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-sqrt.f32 98.4

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

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

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

   5. lift-neg.f32 N/A

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

   6. lower-fma.f32 97.2

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

6. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-lft-identity N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

8. Applied rewrites 97.3%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-/.f32 N/A

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

    2. mul-1-neg N/A

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

    3. lift-neg.f32 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f32 95.3

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

11. Applied rewrites 95.3%

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

Alternative 14: 92.8% accurate, 7.8× 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 97.8%

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

2. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-sqrt.f32 92.8

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

4. Applied rewrites 92.8%

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

Alternative 15: 5.0% accurate, 10.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta, c)
use fmin_fmax_functions
    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 97.8%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f32 5.0

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

4. Applied rewrites 5.0%

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

Reproduce

herbie shell --seed 2025140 
(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))))))