HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 16.4s

Alternatives: 8

Speedup: 2.0×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot \left({\left(e^{\frac{-1}{v}}\right)}^{\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)} \cdot e^{0.6931}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (* (pow (exp (/ -1.0 v)) (fma sinTheta_O sinTheta_i 1.0)) (exp 0.6931))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * (powf(expf((-1.0f / v)), fmaf(sinTheta_O, sinTheta_i, 1.0f)) * expf(0.6931f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * Float32((exp(Float32(Float32(-1.0) / v)) ^ fma(sinTheta_O, sinTheta_i, Float32(1.0))) * exp(Float32(0.6931))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   9. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   10. distribute-neg-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   11. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right)} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \]

   19. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 + \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) \cdot \frac{-1}{v}}} \]
6. Step-by-step derivation

7. Applied rewrites 99.7%

   \[\leadsto \frac{0.5}{v} \cdot \left(e^{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}} \cdot \color{blue}{e^{0.6931}}\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

   \[\leadsto \frac{0.5}{v} \cdot \left({\left(e^{\frac{-1}{v}}\right)}^{\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)\right)} \cdot e^{\color{blue}{0.6931}}\right) \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{\mathsf{fma}\left(v, 0.6931, \mathsf{fma}\left(sinTheta\_O, -sinTheta\_i, -1\right)\right)}{v} + \log \left(\frac{0.5}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (/ (fma v 0.6931 (fma sinTheta_O (- sinTheta_i) -1.0)) v)
   (log (/ 0.5 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((fmaf(v, 0.6931f, fmaf(sinTheta_O, -sinTheta_i, -1.0f)) / v) + logf((0.5f / v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(fma(v, Float32(0.6931), fma(sinTheta_O, Float32(-sinTheta_i), Float32(-1.0))) / v) + log(Float32(Float32(0.5) / v))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   4. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   5. distribute-neg-in N/A

      \[\leadsto e^{\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   7. associate-*r/ N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   8. *-commutative N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   9. associate-/l* N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   10. metadata-eval N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   11. distribute-neg-frac N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   12. distribute-rgt1-in N/A

       \[\leadsto e^{\left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   13. +-commutative N/A

       \[\leadsto e^{\left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   14. lower-fma.f32 N/A

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(1 + sinTheta\_O \cdot sinTheta\_i, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right)} + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   15. +-commutative N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   16. lower-fma.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   18. metadata-eval N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{\color{blue}{-1}}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   19. lower-/.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{-1}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931\right) + \log \left(\frac{0.5}{v}\right)}} \]

6. Taylor expanded in v around 0

   \[\leadsto e^{\frac{-1 \cdot \left(1 + sinTheta\_O \cdot sinTheta\_i\right) + \frac{6931}{10000} \cdot v}{v} + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 99.8%

   \[\leadsto e^{\frac{\mathsf{fma}\left(v, 0.6931, \mathsf{fma}\left(sinTheta\_O, -sinTheta\_i, -1\right)\right)}{v} + \log \color{blue}{\left(\frac{0.5}{v}\right)}} \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931 - \log \left(v \cdot 2\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (fma (fma sinTheta_O sinTheta_i 1.0) (/ -1.0 v) (- 0.6931 (log (* v 2.0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(fmaf(fmaf(sinTheta_O, sinTheta_i, 1.0f), (-1.0f / v), (0.6931f - logf((v * 2.0f)))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(fma(fma(sinTheta_O, sinTheta_i, Float32(1.0)), Float32(Float32(-1.0) / v), Float32(Float32(0.6931) - log(Float32(v * Float32(2.0))))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   4. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   5. distribute-neg-in N/A

      \[\leadsto e^{\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   7. associate-*r/ N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   8. *-commutative N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   9. associate-/l* N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   10. metadata-eval N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   11. distribute-neg-frac N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   12. distribute-rgt1-in N/A

       \[\leadsto e^{\left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   13. +-commutative N/A

       \[\leadsto e^{\left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   14. lower-fma.f32 N/A

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(1 + sinTheta\_O \cdot sinTheta\_i, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right)} + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   15. +-commutative N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   16. lower-fma.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   18. metadata-eval N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{\color{blue}{-1}}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   19. lower-/.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{-1}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931\right) + \log \left(\frac{0.5}{v}\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 99.8%

   \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{-1}{v}}, 0.6931 - \log \left(v \cdot 2\right)\right)} \]
8. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (fma (fma sinTheta_O sinTheta_i 1.0) (/ -1.0 v) 0.6931))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf(fmaf(fmaf(sinTheta_O, sinTheta_i, 1.0f), (-1.0f / v), 0.6931f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(fma(fma(sinTheta_O, sinTheta_i, Float32(1.0)), Float32(Float32(-1.0) / v), Float32(0.6931))))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   9. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   10. distribute-neg-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   11. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right)} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \]

   19. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 + \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) \cdot \frac{-1}{v}}} \]
6. Step-by-step derivation

7. Applied rewrites 99.8%

   \[\leadsto \frac{0.5}{v} \cdot e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931\right)} \]
8. Add Preprocessing

Alternative 5: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (exp (- 0.6931 (/ (fma sinTheta_O sinTheta_i 1.0) v)))) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * expf((0.6931f - (fmaf(sinTheta_O, sinTheta_i, 1.0f) / v)))) / v;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * exp(Float32(Float32(0.6931) - Float32(fma(sinTheta_O, sinTheta_i, Float32(1.0)) / v)))) / v)
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   5. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)\right) + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)}} \]

   6. exp-sum N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v \cdot 2} \cdot e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v}}} \]

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}}}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(e^{\frac{6931}{10000}} \cdot e^{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}}\right)}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(e^{\frac{6931}{10000}} \cdot e^{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}}\right)}{v}} \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}}{v}} \]
8. Add Preprocessing

Alternative 6: 99.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f + (-1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   9. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   10. distribute-neg-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   11. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right)} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \]

   19. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 + \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) \cdot \frac{-1}{v}}} \]

6. Taylor expanded in sinTheta_O around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \frac{-1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.5%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
9. Add Preprocessing

Alternative 7: 97.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ e^{-\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (- (/ (fma sinTheta_O sinTheta_i 1.0) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(-(fmaf(sinTheta_O, sinTheta_i, 1.0f) / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(-Float32(fma(sinTheta_O, sinTheta_i, Float32(1.0)) / v)))
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   4. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   5. distribute-neg-in N/A

      \[\leadsto e^{\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   7. associate-*r/ N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   8. *-commutative N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   9. associate-/l* N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   10. metadata-eval N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   11. distribute-neg-frac N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   12. distribute-rgt1-in N/A

       \[\leadsto e^{\left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   13. +-commutative N/A

       \[\leadsto e^{\left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   14. lower-fma.f32 N/A

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(1 + sinTheta\_O \cdot sinTheta\_i, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right)} + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   15. +-commutative N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   16. lower-fma.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   18. metadata-eval N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{\color{blue}{-1}}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   19. lower-/.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{-1}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931\right) + \log \left(\frac{0.5}{v}\right)}} \]

6. Taylor expanded in v around 0

   \[\leadsto e^{-1 \cdot \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]
7. Step-by-step derivation

8. Applied rewrites 98.0%

   \[\leadsto e^{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\color{blue}{-v}}} \]

9. Final simplification 98.0%

   \[\leadsto e^{-\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \]
10. Add Preprocessing

Alternative 8: 97.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ -1.0 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((-1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((-1.0e0) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(-1.0) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(-1.0) / v));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   4. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   5. distribute-neg-in N/A

      \[\leadsto e^{\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   6. mul-1-neg N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   7. associate-*r/ N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   8. *-commutative N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   9. associate-/l* N/A

      \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   10. metadata-eval N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   11. distribute-neg-frac N/A

       \[\leadsto e^{\left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   12. distribute-rgt1-in N/A

       \[\leadsto e^{\left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   13. +-commutative N/A

       \[\leadsto e^{\left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   14. lower-fma.f32 N/A

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(1 + sinTheta\_O \cdot sinTheta\_i, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right)} + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   15. +-commutative N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   16. lower-fma.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}, \mathsf{neg}\left(\frac{1}{v}\right), \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   18. metadata-eval N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{\color{blue}{-1}}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

   19. lower-/.f32 N/A

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \color{blue}{\frac{-1}{v}}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, 0.6931\right) + \log \left(\frac{0.5}{v}\right)}} \]

6. Taylor expanded in v around 0

   \[\leadsto e^{-1 \cdot \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]
7. Step-by-step derivation

8. Applied rewrites 98.0%

   \[\leadsto e^{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\color{blue}{-v}}} \]

9. Taylor expanded in sinTheta_O around 0

   \[\leadsto e^{\frac{-1}{v}} \]
10. Step-by-step derivation

11. Applied rewrites 98.0%

    \[\leadsto e^{\frac{-1}{v}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))