Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.3%

Time: 21.2s

Alternatives: 8

Speedup: N/A×

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: 98.3% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(sqrt(fma(Float32(-2.0), cosTheta, Float32(1.0))) / cosTheta) / sqrt(Float32(pi))) * Float32(cosh(Float32(cosTheta * cosTheta)) - sinh(Float32(cosTheta * 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   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-PI.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   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

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

4. Applied rewrites 98.1%

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

   1. lift-exp.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. sinh-+-cosh-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lower-cosh.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lower-sinh.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-neg.f32 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}}{cosTheta}}{\sqrt{\pi}} \cdot \left(\cosh \left(cosTheta \cdot cosTheta\right) - \sinh \left(cosTheta \cdot cosTheta\right)\right)} \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(sqrt(fma(Float32(-2.0), cosTheta, Float32(1.0))) / cosTheta) / sqrt(Float32(pi))) * exp(Float32(Float32(-cosTheta) * 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   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-PI.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   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

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 3: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta}\\ t_1 := \mathsf{fma}\left(\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}, t\_0, 1\right)\\ \mathsf{fma}\left(\frac{c}{{t\_1}^{3}} - {t\_1}^{-2}, c, \frac{1}{\mathsf{fma}\left(\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}, t\_0, 1\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (/ (exp (* (- cosTheta) cosTheta)) cosTheta))
        (t_1 (fma (sqrt (/ (fma -2.0 cosTheta 1.0) PI)) t_0 1.0)))
   (fma
    (- (/ c (pow t_1 3.0)) (pow t_1 -2.0))
    c
    (/
     1.0
     (fma
      (sqrt (* cosTheta (- (/ 1.0 (* cosTheta PI)) (* 2.0 (/ 1.0 PI)))))
      t_0
      1.0)))))

C

float code(float cosTheta, float c) {
	float t_0 = expf((-cosTheta * cosTheta)) / cosTheta;
	float t_1 = fmaf(sqrtf((fmaf(-2.0f, cosTheta, 1.0f) / ((float) M_PI))), t_0, 1.0f);
	return fmaf(((c / powf(t_1, 3.0f)) - powf(t_1, -2.0f)), c, (1.0f / fmaf(sqrtf((cosTheta * ((1.0f / (cosTheta * ((float) M_PI))) - (2.0f * (1.0f / ((float) M_PI)))))), t_0, 1.0f)));
}

Julia

function code(cosTheta, c)
	t_0 = Float32(exp(Float32(Float32(-cosTheta) * cosTheta)) / cosTheta)
	t_1 = fma(sqrt(Float32(fma(Float32(-2.0), cosTheta, Float32(1.0)) / Float32(pi))), t_0, Float32(1.0))
	return fma(Float32(Float32(c / (t_1 ^ Float32(3.0))) - (t_1 ^ Float32(-2.0))), c, Float32(Float32(1.0) / fma(sqrt(Float32(cosTheta * Float32(Float32(Float32(1.0) / Float32(cosTheta * Float32(pi))) - Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(pi)))))), t_0, 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. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in cosTheta around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 97.8

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

8. Applied rewrites 97.8%

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

Alternative 4: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{\frac{\mathsf{fma}\left(-2, cosTheta, 1\right)}{\pi}}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta}, 1\right)\\ \mathsf{fma}\left(\frac{c}{{t\_0}^{3}} - {\left(\frac{1}{cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)}^{-2}, c, \frac{1}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0
         (fma
          (sqrt (/ (fma -2.0 cosTheta 1.0) PI))
          (/ (exp (* (- cosTheta) cosTheta)) cosTheta)
          1.0)))
   (fma
    (- (/ c (pow t_0 3.0)) (pow (* (/ 1.0 cosTheta) (/ 1.0 (sqrt PI))) -2.0))
    c
    (/ 1.0 t_0))))

C

float code(float cosTheta, float c) {
	float t_0 = fmaf(sqrtf((fmaf(-2.0f, cosTheta, 1.0f) / ((float) M_PI))), (expf((-cosTheta * cosTheta)) / cosTheta), 1.0f);
	return fmaf(((c / powf(t_0, 3.0f)) - powf(((1.0f / cosTheta) * (1.0f / sqrtf(((float) M_PI)))), -2.0f)), c, (1.0f / t_0));
}

Julia

function code(cosTheta, c)
	t_0 = fma(sqrt(Float32(fma(Float32(-2.0), cosTheta, Float32(1.0)) / Float32(pi))), Float32(exp(Float32(Float32(-cosTheta) * cosTheta)) / cosTheta), Float32(1.0))
	return fma(Float32(Float32(c / (t_0 ^ Float32(3.0))) - (Float32(Float32(Float32(1.0) / cosTheta) * Float32(Float32(1.0) / sqrt(Float32(pi)))) ^ Float32(-2.0))), c, Float32(Float32(1.0) / t_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. Add Preprocessing

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. lift-sqrt.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-/.f32 97.5

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

8. Applied rewrites 97.5%

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

Alternative 5: 95.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 96.1%

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

Alternative 6: 95.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in c around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 95.8%

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

9. Final simplification 95.8%

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

Alternative 7: 80.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{{\left(e^{-1}\right)}^{\left(cosTheta \cdot cosTheta\right)}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\\ \mathbf{if}\;c \leq -5.000000156871975 \cdot 10^{-23} \lor \neg \left(c \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;\left(c \cdot c\right) \cdot \left(\left(\frac{1}{\left(c \cdot c\right) \cdot t\_0} + {t\_0}^{-3}\right) - \frac{1}{c \cdot {t\_0}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta \cdot cosTheta\right) \cdot \frac{\sqrt{\pi} - cosTheta \cdot \left(\pi \cdot \left(1 + \left(c - \frac{1}{\sqrt{\pi}}\right)\right)\right)}{cosTheta}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0
         (+
          1.0
          (*
           (/ (pow (exp -1.0) (* cosTheta cosTheta)) cosTheta)
           (sqrt (/ (+ 1.0 (* -2.0 cosTheta)) PI))))))
   (if (or (<= c -5.000000156871975e-23) (not (<= c 1.9999999996399175e-23)))
     (*
      (* c c)
      (-
       (+ (/ 1.0 (* (* c c) t_0)) (pow t_0 -3.0))
       (/ 1.0 (* c (pow t_0 2.0)))))
     (*
      (* cosTheta cosTheta)
      (/
       (- (sqrt PI) (* cosTheta (* PI (+ 1.0 (- c (/ 1.0 (sqrt PI)))))))
       cosTheta)))))

C

float code(float cosTheta, float c) {
	float t_0 = 1.0f + ((powf(expf(-1.0f), (cosTheta * cosTheta)) / cosTheta) * sqrtf(((1.0f + (-2.0f * cosTheta)) / ((float) M_PI))));
	float tmp;
	if ((c <= -5.000000156871975e-23f) || !(c <= 1.9999999996399175e-23f)) {
		tmp = (c * c) * (((1.0f / ((c * c) * t_0)) + powf(t_0, -3.0f)) - (1.0f / (c * powf(t_0, 2.0f))));
	} else {
		tmp = (cosTheta * cosTheta) * ((sqrtf(((float) M_PI)) - (cosTheta * (((float) M_PI) * (1.0f + (c - (1.0f / sqrtf(((float) M_PI)))))))) / cosTheta);
	}
	return tmp;
}

Julia

function code(cosTheta, c)
	t_0 = Float32(Float32(1.0) + Float32(Float32((exp(Float32(-1.0)) ^ Float32(cosTheta * cosTheta)) / cosTheta) * sqrt(Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * cosTheta)) / Float32(pi)))))
	tmp = Float32(0.0)
	if ((c <= Float32(-5.000000156871975e-23)) || !(c <= Float32(1.9999999996399175e-23)))
		tmp = Float32(Float32(c * c) * Float32(Float32(Float32(Float32(1.0) / Float32(Float32(c * c) * t_0)) + (t_0 ^ Float32(-3.0))) - Float32(Float32(1.0) / Float32(c * (t_0 ^ Float32(2.0))))));
	else
		tmp = Float32(Float32(cosTheta * cosTheta) * Float32(Float32(sqrt(Float32(pi)) - Float32(cosTheta * Float32(Float32(pi) * Float32(Float32(1.0) + Float32(c - Float32(Float32(1.0) / sqrt(Float32(pi)))))))) / cosTheta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta, c)
	t_0 = single(1.0) + (((exp(single(-1.0)) ^ (cosTheta * cosTheta)) / cosTheta) * sqrt(((single(1.0) + (single(-2.0) * cosTheta)) / single(pi))));
	tmp = single(0.0);
	if ((c <= single(-5.000000156871975e-23)) || ~((c <= single(1.9999999996399175e-23))))
		tmp = (c * c) * (((single(1.0) / ((c * c) * t_0)) + (t_0 ^ single(-3.0))) - (single(1.0) / (c * (t_0 ^ single(2.0)))));
	else
		tmp = (cosTheta * cosTheta) * ((sqrt(single(pi)) - (cosTheta * (single(pi) * (single(1.0) + (c - (single(1.0) / sqrt(single(pi)))))))) / cosTheta);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if c < -5.00000016e-23 or 2e-23 < c

   1. Initial program 98.0%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 97.2%

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

   if -5.00000016e-23 < c < 2e-23

   1. Initial program 97.5%

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

   3. Taylor expanded in cosTheta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in cosTheta around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in cosTheta around 0

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

       1. lower-/.f32 N/A

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

   11. Applied rewrites 49.3%

       \[\leadsto \left(cosTheta \cdot cosTheta\right) \cdot \frac{\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \frac{1}{\sqrt{\pi}}\right)\right)\right)\right)}{cosTheta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.000000156871975 \cdot 10^{-23} \lor \neg \left(c \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;\left(c \cdot c\right) \cdot \left(\left(\frac{1}{\left(c \cdot c\right) \cdot \left(1 + \frac{{\left(e^{-1}\right)}^{\left(cosTheta \cdot cosTheta\right)}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)} + {\left(1 + \frac{{\left(e^{-1}\right)}^{\left(cosTheta \cdot cosTheta\right)}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}^{-3}\right) - \frac{1}{c \cdot {\left(1 + \frac{{\left(e^{-1}\right)}^{\left(cosTheta \cdot cosTheta\right)}}{cosTheta} \cdot \sqrt{\frac{1 + -2 \cdot cosTheta}{\pi}}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta \cdot cosTheta\right) \cdot \frac{\sqrt{\pi} - cosTheta \cdot \left(\pi \cdot \left(1 + \left(c - \frac{1}{\sqrt{\pi}}\right)\right)\right)}{cosTheta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = (cosTheta * cosTheta) * ((sqrt(single(pi)) - (cosTheta * (single(pi) * (single(1.0) + (c - (single(1.0) / 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. Add Preprocessing

3. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in cosTheta around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

8. Applied rewrites 54.8%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-/.f32 N/A

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

11. Applied rewrites 54.9%

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

12. Final simplification 54.9%

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

Reproduce

herbie shell --seed 2025057 
(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))))))