HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.4s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0 + (v * log((u + ((1.0e0 - u) * exp(((-2.0e0) / v))))))
end function

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (exp (/ -2.0 v)) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf(((expf((-2.0f / v)) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log(((exp(((-2.0e0) / v)) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log(((exp((single(-2.0) / v)) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Final simplification 99.5%

   \[\leadsto \log \left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
4. Add Preprocessing

Alternative 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(\left(--1\right) - e^{\frac{-2}{v}}\right) \cdot u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (* (- (- -1.0) (exp (/ -2.0 v))) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf(((-(-1.0f) - expf((-2.0f / v))) * u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log(((-(-1.0e0) - exp(((-2.0e0) / v))) * u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(-Float32(-1.0)) - exp(Float32(Float32(-2.0) / v))) * u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log(((-single(-1.0) - exp((single(-2.0) / v))) * u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around inf

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

   1. *-commutative N/A

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(\left(1 + -1 \cdot e^{\frac{-2}{v}}\right) \cdot u\right)} \]

   2. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(\color{blue}{\left(-1 \cdot e^{\frac{-2}{v}} + 1\right)} \cdot u\right) \]

   3. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(\left(-1 \cdot e^{\frac{-2}{v}} + \color{blue}{-1 \cdot -1}\right) \cdot u\right) \]

   4. distribute-lft-in N/A

      \[\leadsto 1 + v \cdot \log \left(\color{blue}{\left(-1 \cdot \left(e^{\frac{-2}{v}} + -1\right)\right)} \cdot u\right) \]

   5. metadata-eval N/A

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

   6. sub-neg N/A

      \[\leadsto 1 + v \cdot \log \left(\left(-1 \cdot \color{blue}{\left(e^{\frac{-2}{v}} - 1\right)}\right) \cdot u\right) \]

   7. associate-*r* N/A

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(-1 \cdot \left(\left(e^{\frac{-2}{v}} - 1\right) \cdot u\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(-1 \cdot \color{blue}{\left(u \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)}\right) \]

   9. associate-*r* N/A

      \[\leadsto 1 + v \cdot \log \color{blue}{\left(\left(-1 \cdot u\right) \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto 1 + v \cdot \log \color{blue}{\left(\left(-1 \cdot u\right) \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)} \]

   11. neg-mul-1 N/A

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

   12. lower-neg.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(\color{blue}{\left(-u\right)} \cdot \left(e^{\frac{-2}{v}} - 1\right)\right) \]

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

   17. lower-expm1.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)}\right) \]

   18. associate-*r/ N/A

       \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{v}}\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\mathsf{neg}\left(\frac{\color{blue}{2}}{v}\right)\right)\right) \]

   20. distribute-neg-frac N/A

       \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{\mathsf{neg}\left(2\right)}{v}}\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{\color{blue}{-2}}{v}\right)\right) \]

   22. lower-/.f32 45.6

       \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \mathsf{expm1}\left(\color{blue}{\frac{-2}{v}}\right)\right) \]

5. Applied rewrites 46.0%

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(\left(-u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.4%

   \[\leadsto 1 + v \cdot \log \left(\left(-u\right) \cdot \left(e^{\frac{-2}{v}} - \color{blue}{1}\right)\right) \]

8. Final simplification 94.4%

   \[\leadsto \log \left(\left(\left(--1\right) - e^{\frac{-2}{v}}\right) \cdot u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 3: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  (*
   (log
    (+
     (*
      (/ 1.0 (- 1.0 (/ (- (/ (+ (/ -1.3333333333333333 v) -2.0) v) 2.0) v)))
      (- 1.0 u))
     u))
   v)
  1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / (1.0f - (((((-1.3333333333333333f / v) + -2.0f) / v) - 2.0f) / v))) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / (1.0e0 - ((((((-1.3333333333333333e0) / v) + (-2.0e0)) / v) - 2.0e0) / v))) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(Float32(-1.3333333333333333) / v) + Float32(-2.0)) / v) - Float32(2.0)) / v))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / (single(1.0) - (((((single(-1.3333333333333333) / v) + single(-2.0)) / v) - single(2.0)) / v))) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{-2}{v}}}\right) \]

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.4

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{e^{\frac{\color{blue}{2}}{v}}}\right) \]

4. Applied rewrites 99.4%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{e^{\frac{2}{v}}}}\right) \]

5. Taylor expanded in v around -inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + -1 \cdot \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)}}\right) \]

   2. unsub-neg N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

   3. lower--.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{1 - \color{blue}{\frac{-1 \cdot \frac{2 + \frac{4}{3} \cdot \frac{1}{v}}{v} - 2}{v}}}\right) \]

7. Applied rewrites 93.9%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}}}\right) \]

8. Final simplification 93.9%

   \[\leadsto \log \left(\frac{1}{1 - \frac{\frac{\frac{-1.3333333333333333}{v} + -2}{v} - 2}{v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 4: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\left(\frac{2}{v} + 1\right) + \frac{2}{v \cdot v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  (* (log (+ (* (/ 1.0 (+ (+ (/ 2.0 v) 1.0) (/ 2.0 (* v v)))) (- 1.0 u)) u)) v)
  1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / (((2.0f / v) + 1.0f) + (2.0f / (v * v)))) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / (((2.0e0 / v) + 1.0e0) + (2.0e0 / (v * v)))) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(2.0) / v) + Float32(1.0)) + Float32(Float32(2.0) / Float32(v * v)))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / (((single(2.0) / v) + single(1.0)) + (single(2.0) / (v * v)))) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{-2}{v}}}\right) \]

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.4

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{e^{\frac{\color{blue}{2}}{v}}}\right) \]

4. Applied rewrites 99.4%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{e^{\frac{2}{v}}}}\right) \]

5. Taylor expanded in v around inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + \left(2 \cdot \frac{1}{v} + \frac{2}{{v}^{2}}\right)}}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\left(2 \cdot \frac{1}{v} + \frac{2}{{v}^{2}}\right) + 1}}\right) \]

   2. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\left(2 \cdot \frac{1}{v} + \frac{\color{blue}{2 \cdot 1}}{{v}^{2}}\right) + 1}\right) \]

   3. associate-*r/ N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\left(2 \cdot \frac{1}{v} + \color{blue}{2 \cdot \frac{1}{{v}^{2}}}\right) + 1}\right) \]

   4. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\left(2 \cdot \frac{1}{{v}^{2}} + 2 \cdot \frac{1}{v}\right)} + 1}\right) \]

   5. associate-+l+ N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{2 \cdot \frac{1}{{v}^{2}} + \left(2 \cdot \frac{1}{v} + 1\right)}}\right) \]

   6. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{2 \cdot \frac{1}{{v}^{2}} + \color{blue}{\left(1 + 2 \cdot \frac{1}{v}\right)}}\right) \]

   7. lower-+.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{2 \cdot \frac{1}{{v}^{2}} + \left(1 + 2 \cdot \frac{1}{v}\right)}}\right) \]

   8. associate-*r/ N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2 \cdot 1}{{v}^{2}}} + \left(1 + 2 \cdot \frac{1}{v}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{\color{blue}{2}}{{v}^{2}} + \left(1 + 2 \cdot \frac{1}{v}\right)}\right) \]

   10. lower-/.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2}{{v}^{2}}} + \left(1 + 2 \cdot \frac{1}{v}\right)}\right) \]

   11. unpow2 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{\color{blue}{v \cdot v}} + \left(1 + 2 \cdot \frac{1}{v}\right)}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{\color{blue}{v \cdot v}} + \left(1 + 2 \cdot \frac{1}{v}\right)}\right) \]

   13. +-commutative N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{v \cdot v} + \color{blue}{\left(2 \cdot \frac{1}{v} + 1\right)}}\right) \]

   14. lower-+.f32 N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{v \cdot v} + \color{blue}{\left(2 \cdot \frac{1}{v} + 1\right)}}\right) \]

   15. associate-*r/ N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{v \cdot v} + \left(\color{blue}{\frac{2 \cdot 1}{v}} + 1\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{v \cdot v} + \left(\frac{\color{blue}{2}}{v} + 1\right)}\right) \]

   17. lower-/.f32 91.8

       \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{2}{v \cdot v} + \left(\color{blue}{\frac{2}{v}} + 1\right)}\right) \]

7. Applied rewrites 91.8%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2}{v \cdot v} + \left(\frac{2}{v} + 1\right)}}\right) \]

8. Final simplification 91.8%

   \[\leadsto \log \left(\frac{1}{\left(\frac{2}{v} + 1\right) + \frac{2}{v \cdot v}} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 5: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\frac{2}{v} + 1} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ (* (log (+ (* (/ 1.0 (+ (/ 2.0 v) 1.0)) (- 1.0 u)) u)) v) 1.0))

C

float code(float u, float v) {
	return (logf((((1.0f / ((2.0f / v) + 1.0f)) * (1.0f - u)) + u)) * v) + 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = (log((((1.0e0 / ((2.0e0 / v) + 1.0e0)) * (1.0e0 - u)) + u)) * v) + 1.0e0
end function

Julia

function code(u, v)
	return Float32(Float32(log(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / v) + Float32(1.0))) * Float32(Float32(1.0) - u)) + u)) * v) + Float32(1.0))
end

MATLAB

function tmp = code(u, v)
	tmp = (log((((single(1.0) / ((single(2.0) / v) + single(1.0))) * (single(1.0) - u)) + u)) * v) + single(1.0);
end

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{e^{\frac{-2}{v}}}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{\frac{-2}{v}}}\right) \]

   3. frac-2neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 99.4

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{e^{\frac{\color{blue}{2}}{v}}}\right) \]

4. Applied rewrites 99.4%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\frac{1}{e^{\frac{2}{v}}}}\right) \]

5. Taylor expanded in v around inf

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{1 + 2 \cdot \frac{1}{v}}}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{2 \cdot \frac{1}{v} + 1}}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{2 \cdot \frac{1}{v} + 1}}\right) \]

   3. associate-*r/ N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2 \cdot 1}{v}} + 1}\right) \]

   4. metadata-eval N/A

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\frac{\color{blue}{2}}{v} + 1}\right) \]

   5. lower-/.f32 88.8

      \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2}{v}} + 1}\right) \]

7. Applied rewrites 88.8%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \frac{1}{\color{blue}{\frac{2}{v} + 1}}\right) \]

8. Final simplification 88.8%

   \[\leadsto \log \left(\frac{1}{\frac{2}{v} + 1} \cdot \left(1 - u\right) + u\right) \cdot v + 1 \]
9. Add Preprocessing

Alternative 6: 90.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u \cdot u\right) \cdot \left(\frac{2}{v} - \frac{\left(\frac{2}{v} + 2\right) - \frac{2}{u}}{u}\right)}{-v} \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    (*
     (/ (* (* u u) (- (/ 2.0 v) (/ (- (+ (/ 2.0 v) 2.0) (/ 2.0 u)) u))) (- v))
     v)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = ((((u * u) * ((2.0f / v) - ((((2.0f / v) + 2.0f) - (2.0f / u)) / u))) / -v) * v) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = ((((u * u) * ((2.0e0 / v) - ((((2.0e0 / v) + 2.0e0) - (2.0e0 / u)) / u))) / -v) * v) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(u * u) * Float32(Float32(Float32(2.0) / v) - Float32(Float32(Float32(Float32(Float32(2.0) / v) + Float32(2.0)) - Float32(Float32(2.0) / u)) / u))) / Float32(-v)) * v) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = ((((u * u) * ((single(2.0) / v) - ((((single(2.0) / v) + single(2.0)) - (single(2.0) / u)) / u))) / -v) * v) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.1%

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

   if 0.150000006 < v

   1. Initial program 93.4%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + v \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 1 + v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v} + 2 \cdot \left(1 - u\right)}{v}\right)\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 1 + v \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v} + 2 \cdot \left(1 - u\right)}{\mathsf{neg}\left(v\right)}} \]

      3. lower-/.f32 N/A

         \[\leadsto 1 + v \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v} + 2 \cdot \left(1 - u\right)}{\mathsf{neg}\left(v\right)}} \]

   5. Applied rewrites 6.9%

      \[\leadsto 1 + v \cdot \color{blue}{\frac{\mathsf{fma}\left(2, 1 - u, \frac{\mathsf{fma}\left({\left(1 - u\right)}^{2}, 2, \left(1 - u\right) \cdot -2\right)}{v}\right)}{-v}} \]

   6. Taylor expanded in u around -inf

      \[\leadsto 1 + v \cdot \frac{{u}^{2} \cdot \left(-1 \cdot \frac{\left(2 + 4 \cdot \frac{1}{v}\right) - \left(2 \cdot \frac{1}{v} + 2 \cdot \frac{1}{u}\right)}{u} + 2 \cdot \frac{1}{v}\right)}{-\color{blue}{v}} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.4%

      \[\leadsto 1 + v \cdot \frac{\left(\frac{2}{v} - \frac{\left(\frac{2}{v} + 2\right) - \frac{2}{u}}{u}\right) \cdot \left(u \cdot u\right)}{-\color{blue}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u \cdot u\right) \cdot \left(\frac{2}{v} - \frac{\left(\frac{2}{v} + 2\right) - \frac{2}{u}}{u}\right)}{-v} \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 231.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

float code(float u, float v) {
	return 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

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

5. Applied rewrites 84.7%

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

Alternative 8: 6.0% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.7%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))