HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 8.3s

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -1:\\ \;\;\;\;1 + \left(\left(2 \cdot u - \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{v}\right) - 2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= (* (log (- u (* (+ -1.0 u) (exp (/ -2.0 v))))) v) -1.0)
   (+
    1.0
    (-
     (-
      (* 2.0 u)
      (/
       (-
        (* -2.0 u)
        (/ (fma 0.6666666666666666 (/ u v) (* 1.3333333333333333 u)) v))
       v))
     2.0))
   1.0))

C

float code(float u, float v) {
	float tmp;
	if ((logf((u - ((-1.0f + u) * expf((-2.0f / v))))) * v) <= -1.0f) {
		tmp = 1.0f + (((2.0f * u) - (((-2.0f * u) - (fmaf(0.6666666666666666f, (u / v), (1.3333333333333333f * u)) / v)) / v)) - 2.0f);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (Float32(log(Float32(u - Float32(Float32(Float32(-1.0) + u) * exp(Float32(Float32(-2.0) / v))))) * v) <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(2.0) * u) - Float32(Float32(Float32(Float32(-2.0) * u) - Float32(fma(Float32(0.6666666666666666), Float32(u / v), Float32(Float32(1.3333333333333333) * u)) / v)) / v)) - Float32(2.0)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))) < -1

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. rec-exp N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-expm1.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 32.7

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 66.0%

      \[\leadsto 1 + \left(\left(2 \cdot u - \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{v}\right) - \color{blue}{2}\right) \]

   if -1 < (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))

   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 93.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(u - \left(-1 + u\right) \cdot e^{\frac{-2}{v}}\right) \cdot v \leq -1:\\ \;\;\;\;1 + \left(\left(2 \cdot u - \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{v}\right) - 2\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower--.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 2.4

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

5. Applied rewrites 2.4%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{\left(1 - \frac{2 - \frac{2 - \frac{1.3333333333333333}{v}}{v}}{v}\right)}\right) \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. *-lft-identity N/A

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

   3. lower-fma.f32 20.6

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 22.7

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

7. Applied rewrites 39.5%

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

8. Taylor expanded in u around inf

   \[\leadsto 1 + v \cdot \log \color{blue}{\left(u \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)} \]
9. 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. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-exp.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 95.0

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

10. Applied rewrites 95.0%

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

11. Final simplification 95.0%

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

Alternative 4: 87.3% 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 99.6%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 88.7%

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

Alternative 5: 5.7% 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 99.6%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 5.6%

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

Reproduce

herbie shell --seed 2024332 
(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))))))))