HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.8s

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

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

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

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. Add Preprocessing

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (* (expm1 (/ -2.0 v)) (- u))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-u)))))
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 inf

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\color{blue}{\left(u \cdot \left(1 + -1 \cdot e^{\frac{-2}{v}}\right)\right)}\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(-1 \cdot e^{\frac{-2}{v}} + 1\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\left(\mathsf{neg}\left(e^{\frac{-2}{v}}\right)\right) + 1\right)\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\left(\mathsf{neg}\left(e^{\frac{-2}{v}}\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)\right)\right)\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} + -1\right)\right)\right)\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(\left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\mathsf{neg}\left(u \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(-1 \cdot \left(u \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(-1 \cdot u\right) \cdot \left(e^{\frac{-2}{v}} - 1\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(e^{\frac{-2}{v}} - 1\right) \cdot \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(e^{\frac{-2}{v}} - 1\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   12. expm1-define N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{-2}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\frac{\mathsf{neg}\left(2\right)}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   16. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{expm1}\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   17. expm1-lowering-expm1.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(2 \cdot \frac{1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   18. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(\frac{2 \cdot 1}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\mathsf{neg}\left(\frac{2}{v}\right)\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   20. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{\mathsf{neg}\left(2\right)}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\left(\frac{-2}{v}\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   22. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(-1 \cdot u\right)\right)\right)\right)\right) \]

   23. neg-mul-1 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{*.f32}\left(\mathsf{expm1.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), \left(\mathsf{neg}\left(u\right)\right)\right)\right)\right)\right) \]

5. Simplified 96.3%

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

Alternative 3: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - u\right) \cdot \left(1 - u\right)\\ t_1 := t\_0 - u \cdot u\\ t_2 := 1 + u \cdot -2\\ t_3 := u \cdot u + \left(1 - u\right) \cdot \left(u + -1\right)\\ 1 - v \cdot \log \left(\frac{\frac{\left(1 - u\right) \cdot 2}{t\_3} + \frac{-4}{t\_1} \cdot \frac{t\_0 \cdot t\_2}{t\_3}}{v} + \frac{t\_2}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (* (- 1.0 u) (- 1.0 u)))
        (t_1 (- t_0 (* u u)))
        (t_2 (+ 1.0 (* u -2.0)))
        (t_3 (+ (* u u) (* (- 1.0 u) (+ u -1.0)))))
   (-
    1.0
    (*
     v
     (log
      (+
       (/ (+ (/ (* (- 1.0 u) 2.0) t_3) (* (/ -4.0 t_1) (/ (* t_0 t_2) t_3))) v)
       (/ t_2 t_1)))))))

C

float code(float u, float v) {
	float t_0 = (1.0f - u) * (1.0f - u);
	float t_1 = t_0 - (u * u);
	float t_2 = 1.0f + (u * -2.0f);
	float t_3 = (u * u) + ((1.0f - u) * (u + -1.0f));
	return 1.0f - (v * logf(((((((1.0f - u) * 2.0f) / t_3) + ((-4.0f / t_1) * ((t_0 * t_2) / t_3))) / v) + (t_2 / t_1))));
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    real(4) :: t_3
    t_0 = (1.0e0 - u) * (1.0e0 - u)
    t_1 = t_0 - (u * u)
    t_2 = 1.0e0 + (u * (-2.0e0))
    t_3 = (u * u) + ((1.0e0 - u) * (u + (-1.0e0)))
    code = 1.0e0 - (v * log(((((((1.0e0 - u) * 2.0e0) / t_3) + (((-4.0e0) / t_1) * ((t_0 * t_2) / t_3))) / v) + (t_2 / t_1))))
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))
	t_1 = Float32(t_0 - Float32(u * u))
	t_2 = Float32(Float32(1.0) + Float32(u * Float32(-2.0)))
	t_3 = Float32(Float32(u * u) + Float32(Float32(Float32(1.0) - u) * Float32(u + Float32(-1.0))))
	return Float32(Float32(1.0) - Float32(v * log(Float32(Float32(Float32(Float32(Float32(Float32(Float32(1.0) - u) * Float32(2.0)) / t_3) + Float32(Float32(Float32(-4.0) / t_1) * Float32(Float32(t_0 * t_2) / t_3))) / v) + Float32(t_2 / t_1)))))
end

MATLAB

function tmp = code(u, v)
	t_0 = (single(1.0) - u) * (single(1.0) - u);
	t_1 = t_0 - (u * u);
	t_2 = single(1.0) + (u * single(-2.0));
	t_3 = (u * u) + ((single(1.0) - u) * (u + single(-1.0)));
	tmp = single(1.0) - (v * log(((((((single(1.0) - u) * single(2.0)) / t_3) + ((single(-4.0) / t_1) * ((t_0 * t_2) / t_3))) / v) + (t_2 / t_1))));
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. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u\right)\right)\right)\right) \]

   2. fma-define N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right)\right) \]

   3. fma-lowering-fma.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{fma.f32}\left(\left(1 - u\right), \left(e^{\frac{-2}{v}}\right), u\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{fma.f32}\left(\mathsf{\_.f32}\left(1, u\right), \left(e^{\frac{-2}{v}}\right), u\right)\right)\right)\right) \]

   5. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{fma.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{exp.f32}\left(\left(\frac{-2}{v}\right)\right), u\right)\right)\right)\right) \]

   6. /-lowering-/.f32 99.7%

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{log.f32}\left(\mathsf{fma.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(-2, v\right)\right), u\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u}{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \log \left(\frac{1}{\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u}}\right)\right)\right) \]

   3. log-rec N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \left(\mathsf{neg}\left(\log \left(\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u}\right)\right)\right)\right)\right) \]

   4. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\log \left(\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u}\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u}\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} - u\right), \left(\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) - u \cdot u\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

7. Taylor expanded in v around -inf

   \[\leadsto \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(v, \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\left(-1 \cdot \frac{2 \cdot \frac{1 - u}{{\left(1 - u\right)}^{2} - {u}^{2}} - 4 \cdot \frac{{\left(1 - u\right)}^{2} \cdot \left(1 - 2 \cdot u\right)}{{\left({\left(1 - u\right)}^{2} - {u}^{2}\right)}^{2}}}{v} + \frac{1}{{\left(1 - u\right)}^{2} - {u}^{2}}\right) - 2 \cdot \frac{u}{{\left(1 - u\right)}^{2} - {u}^{2}}\right)}\right)\right)\right)\right) \]

9. Simplified 89.6%

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

10. Final simplification 89.6%

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

Alternative 4: 86.8% accurate, 213.0× 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 88.8%

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

Alternative 5: 5.9% accurate, 213.0× 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 4.8%

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

Reproduce

herbie shell --seed 2024163 
(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))))))))