HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 6.9s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 95.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. sinh-+-cosh-rev N/A

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

   3. flip-+ N/A

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

   4. sinh-cosh N/A

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

   5. sinh---cosh-rev N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. distribute-neg-frac 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. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

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 95.7%

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

8. Final simplification 95.7%

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

Alternative 3: 93.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. sinh-+-cosh-rev N/A

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

   3. flip-+ N/A

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

   4. sinh-cosh N/A

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

   5. sinh---cosh-rev N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. distribute-neg-frac 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. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

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. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 94.7

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

7. Applied rewrites 94.7%

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

8. Final simplification 94.7%

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

Alternative 4: 91.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. sinh-+-cosh-rev N/A

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

   3. flip-+ N/A

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

   4. sinh-cosh N/A

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

   5. sinh---cosh-rev N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. distribute-neg-frac 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. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

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. cancel-sign-sub-inv N/A

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

   2. distribute-lft-neg-in N/A

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

   3. rem-log-exp N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. rec-exp N/A

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

   7. lower--.f32 N/A

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

   8. log-rec N/A

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

   9. rem-log-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 92.6

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

7. Applied rewrites 92.6%

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

8. Final simplification 92.6%

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

Alternative 5: 87.0% 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 99.7%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 89.1%

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

Alternative 6: 5.8% 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 99.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.4%

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

Reproduce

herbie shell --seed 2024312 
(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))))))))