Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 5.7s

Alternatives: 18

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

3. Applied rewrites 98.5%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 98.0%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

Alternative 4: 97.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{{cosTheta}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1\right) + 1}{cosTheta}, c\right) + 1} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1\right) \cdot {cosTheta}^{2} + 1}{cosTheta}, c\right) + 1} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\left(\frac{-1}{6} \cdot {cosTheta}^{2} + \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   8. lower-fma.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, {cosTheta}^{2}, \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   11. pow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta}, c\right) + 1} \]

   13. pow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, cosTheta \cdot cosTheta, 1\right)}{cosTheta}, c\right) + 1} \]

   14. lift-*.f32 97.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, cosTheta \cdot cosTheta, 0.5\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, cosTheta \cdot cosTheta, 1\right)}{cosTheta}, c\right) + 1} \]

7. Applied rewrites 97.8%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, cosTheta \cdot cosTheta, 0.5\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, cosTheta \cdot cosTheta, 1\right)}{cosTheta}, c\right) + 1} \]
8. Add Preprocessing

Alternative 5: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. pow2 N/A

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

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

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

   4. count-2-rev N/A

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

   5. associate--l- N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

7. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f32 N/A

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

9. Applied rewrites 97.6%

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

10. Taylor expanded in cosTheta around 0

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

    1. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \left({cosTheta}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1\right) + 1\right)}{cosTheta} + 1} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \left(\left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1\right) \cdot {cosTheta}^{2} + 1\right)}{cosTheta} + 1} \]

    3. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    4. lower--.f32 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    6. lower-*.f32 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    7. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\left(\frac{-1}{6} \cdot {cosTheta}^{2} + \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    8. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, {cosTheta}^{2}, \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    9. pow2 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    10. lift-*.f32 N/A

        \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot {cosTheta}^{2} - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    11. pow2 N/A

        \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    12. lift-*.f32 N/A

        \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, {cosTheta}^{2}, 1\right)}{cosTheta} + 1} \]

    13. pow2 N/A

        \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, cosTheta \cdot cosTheta, \frac{1}{2}\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, cosTheta \cdot cosTheta, 1\right)}{cosTheta} + 1} \]

    14. lift-*.f32 97.5

        \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, cosTheta \cdot cosTheta, 0.5\right) \cdot \left(cosTheta \cdot cosTheta\right) - 1, cosTheta \cdot cosTheta, 1\right)}{cosTheta} + 1} \]

12. Applied rewrites 97.5%

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

Alternative 6: 97.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 97.4

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

4. Applied rewrites 97.4%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

6. Applied rewrites 97.6%

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

Alternative 7: 97.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift-fma.f32 N/A

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

   11. lift-*.f32 97.5

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

7. Applied rewrites 97.5%

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

Alternative 8: 97.5% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/ (* 1.0 (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(((1.0f * 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(Float32(Float32(1.0) * 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 97.9%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 97.5

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

6. Applied rewrites 97.5%

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

Alternative 9: 97.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (/ (sqrt (/ (- (- 1.0 cosTheta) cosTheta) 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) / ((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(Float32(1.0) - cosTheta) - cosTheta) / Float32(pi))) / cosTheta), fma(Float32(-cosTheta), cosTheta, Float32(1.0)), Float32(c + Float32(1.0))))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 97.4

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

4. Applied rewrites 97.4%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 97.0

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

9. Applied rewrites 97.0%

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

Alternative 10: 96.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 96.9

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

7. Applied rewrites 96.9%

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

Alternative 11: 96.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. pow2 N/A

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

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

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

   4. count-2-rev N/A

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

   5. associate--l- N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

7. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f32 N/A

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

9. Applied rewrites 97.6%

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

10. Taylor expanded in cosTheta around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. pow2 N/A

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

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

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

    5. lower-fma.f32 N/A

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

    6. lift-neg.f32 96.7

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

12. Applied rewrites 96.7%

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

Alternative 12: 96.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. pow2 N/A

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

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

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

   4. count-2-rev N/A

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

   5. associate--l- N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unpow2 N/A

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

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

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

   5. lower-fma.f32 N/A

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

   6. lift-neg.f32 96.6

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

10. Applied rewrites 96.6%

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

Alternative 13: 95.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

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

Alternative 14: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

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

Alternative 15: 95.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. sqrt-div N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-PI.f32 95.6

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

7. Applied rewrites 95.6%

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

Alternative 16: 92.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

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-sqrt.f32 N/A

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

   4. lift-PI.f32 92.8

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

4. Applied rewrites 92.8%

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

Alternative 17: 10.8% accurate, 12.2× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 (/ 1.0 (+ c 1.0)))

C

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

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 + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in c around inf

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

7. Applied rewrites 10.8%

   \[\leadsto \frac{1}{c + 1} \]
8. Add Preprocessing

Alternative 18: 5.0% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

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 97.9%

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

2. Taylor expanded in c around inf

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

4. Applied rewrites 5.0%

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

Reproduce

herbie shell --seed 2025101 
(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))))))