HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 9.5s

Alternatives: 5

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 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 47.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u \cdot u\right) \cdot \left(\left(\frac{\frac{\frac{4}{v}}{v} - \frac{\frac{2}{v} - 2}{v}}{u} - \frac{2}{v \cdot v}\right) - \frac{\frac{\frac{2}{u}}{u}}{v}\right)\right) \cdot v + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ (* (log (fma (exp (/ -2.0 v)) (- 1.0 u) u)) v) 1.0)
   (+
    (*
     (*
      (* u u)
      (-
       (- (/ (- (/ (/ 4.0 v) v) (/ (- (/ 2.0 v) 2.0) v)) u) (/ 2.0 (* v v)))
       (/ (/ (/ 2.0 u) u) v)))
     v)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = (logf(fmaf(expf((-2.0f / v)), (1.0f - u), u)) * v) + 1.0f;
	} else {
		tmp = (((u * u) * ((((((4.0f / v) / v) - (((2.0f / v) - 2.0f) / v)) / u) - (2.0f / (v * v))) - (((2.0f / u) / u) / v))) * v) + 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(log(fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u)) * v) + Float32(1.0));
	else
		tmp = Float32(Float32(Float32(Float32(u * u) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(4.0) / v) / v) - Float32(Float32(Float32(Float32(2.0) / v) - Float32(2.0)) / v)) / u) - Float32(Float32(2.0) / Float32(v * v))) - Float32(Float32(Float32(Float32(2.0) / u) / u) / v))) * v) + Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   4. Applied rewrites 99.4%

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

   if 0.100000001 < v

   1. Initial program 95.5%

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

   3. Taylor expanded in v around -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 9.0%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 50.5%

      \[\leadsto 1 + v \cdot \left(\left(\left(\frac{-\left(\frac{\frac{2}{v} - 2}{v} - \frac{\frac{4}{v}}{v}\right)}{u} - \frac{2}{v \cdot v}\right) - \frac{\frac{\frac{2}{u}}{u}}{v}\right) \cdot \color{blue}{\left(u \cdot u\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot v + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u \cdot u\right) \cdot \left(\left(\frac{\frac{\frac{4}{v}}{v} - \frac{\frac{2}{v} - 2}{v}}{u} - \frac{2}{v \cdot v}\right) - \frac{\frac{\frac{2}{u}}{u}}{v}\right)\right) \cdot v + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% 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 99.7%

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

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

   3. exp-prod N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. associate-/r/ N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. exp-1-e N/A

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

   12. lower-E.f32 N/A

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

   13. neg-mul-1 N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 2.3%

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

8. Taylor expanded in u around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-*.f32 N/A

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in u around inf

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

13. Applied rewrites 94.6%

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

14. Final simplification 94.6%

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

Alternative 4: 86.7% 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 88.8%

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

Alternative 5: 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 5.2%

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

Reproduce

herbie shell --seed 2024276 
(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))))))))