HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.9s

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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(1.0) - u), 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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (exp(((-2.0e0) / v)) * ((-1.0e0) + (1.0e0 / u)))))))
end function

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u * (single(1.0) + (exp((single(-2.0) / v)) * (single(-1.0) + (single(1.0) / 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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in u around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. associate-/l* 99.8%

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

   4. distribute-lft-out 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 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 4: 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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in u around inf 97.2%

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

   1. *-commutative 97.2%

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

   2. +-commutative 97.2%

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

   3. mul-1-neg 97.2%

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

   4. metadata-eval 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. metadata-eval 97.2%

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

   7. sub-neg 97.2%

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

   8. distribute-lft-neg-in 97.2%

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

   9. distribute-rgt-neg-in 97.2%

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

   10. expm1-define 97.2%

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Alternative 5: 87.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf(((u * 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 * 2.0e0) / v)))
end function

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log(((u * 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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in u around inf 97.2%

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

   1. *-commutative 97.2%

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

   2. +-commutative 97.2%

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

   3. mul-1-neg 97.2%

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

   4. metadata-eval 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. metadata-eval 97.2%

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

   7. sub-neg 97.2%

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

   8. distribute-lft-neg-in 97.2%

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

   9. distribute-rgt-neg-in 97.2%

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

   10. expm1-define 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in v around inf 90.1%

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

   1. fma-undefine 90.1%

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

   2. associate-*r/ 90.1%

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

10. Applied egg-rr 90.1%

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

11. Final simplification 90.1%

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

Alternative 6: 87.0% 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. Step-by-step derivation

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around 0 90.0%

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

Alternative 7: 6.0% 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 4.7%

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

Reproduce

herbie shell --seed 2024092 
(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))))))))