HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 11.9s

Alternatives: 7

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: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.7

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

5. Simplified 99.7%

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

   1. flip3-- N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, \color{blue}{\frac{{1}^{3} - {u}^{3}}{1 \cdot 1 + \left(u \cdot u + 1 \cdot u\right)}}, u\right)\right), 1\right) \]

   2. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, \color{blue}{\frac{{1}^{3} - {u}^{3}}{1 \cdot 1 + \left(u \cdot u + 1 \cdot u\right)}}, u\right)\right), 1\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, \frac{\color{blue}{1} - {u}^{3}}{1 \cdot 1 + \left(u \cdot u + 1 \cdot u\right)}, u\right)\right), 1\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, \frac{\color{blue}{1 - {u}^{3}}}{1 \cdot 1 + \left(u \cdot u + 1 \cdot u\right)}, u\right)\right), 1\right) \]

   5. cube-mult N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-+.f32 99.8

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

7. Applied egg-rr 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.7

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

5. Simplified 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-fma.f32 99.7

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

   12. lift-/.f32 99.7

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

   13. lower-fma.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lower-+.f32 99.7

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u)), Float32(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

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.7

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right), 1\right)} \]
6. Add Preprocessing

Alternative 4: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(exp(Float32(Float32(-2.0) / v)) + u)), Float32(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

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower--.f32 99.7

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

5. Simplified 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-fma.f32 99.7

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. lift-/.f32 N/A

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

   10. rec-exp N/A

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

   11. lift-exp.f32 N/A

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

   12. lift-/.f32 99.7

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

   13. lower-fma.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lower-+.f32 99.7

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-exp.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 97.4

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

10. Simplified 97.4%

    \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{e^{\frac{-2}{v}}} + u\right), 1\right) \]
11. Add Preprocessing

Alternative 5: 94.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(v, \log u, 1\right) \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (fma v (log u) 1.0))

C

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

Julia

function code(u, v)
	return fma(v, log(u), Float32(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. frac-2neg N/A

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

   2. distribute-frac-neg2 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1 + v \cdot \log u} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{v \cdot \log u + 1} \]

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. lower-fma.f32 N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. mul-1-neg N/A

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

   11. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(v, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log u\right)\right)}\right), 1\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

   13. lower-log.f32 96.0

       \[\leadsto \mathsf{fma}\left(v, \color{blue}{\log u}, 1\right) \]

7. Simplified 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log u, 1\right)} \]
8. Add Preprocessing

Alternative 6: 86.4% 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. Simplified 89.2%

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

Alternative 7: 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 99.7%

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

3. Taylor expanded in u around 0

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

5. Simplified 5.1%

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

Reproduce

herbie shell --seed 2024214 
(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))))))))