HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 10.2s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(Float32(2.0) - u) + Float32(-1.0)), exp(Float32(Float32(-2.0) / v)), u)), Float32(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. Step-by-step derivation

   1. +-commutative 99.6%

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

   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. expm1-log1p-u 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]
7. Step-by-step derivation

   1. expm1-undefine 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} - 1}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   2. sub-neg 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} + \left(-1\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   3. log1p-undefine 99.8%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

8. Simplified 99.8%

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

Alternative 2: 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 99.6%

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

   1. +-commutative 99.6%

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

   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 3: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(Float32(Float32(Float32(2.0) - u) + Float32(-1.0)) * exp(Float32(Float32(-2.0) / v))))), Float32(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. Step-by-step derivation

   1. +-commutative 99.6%

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

   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 99.7%

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

6. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(v, \log \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)} \cdot e^{\frac{-2}{v}} + u\right), 1\right) \]
9. Step-by-step derivation

   1. expm1-undefine 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} - 1}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   2. sub-neg 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} + \left(-1\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   3. log1p-undefine 99.8%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 0.7× 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 99.6%

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

   1. +-commutative 99.6%

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

   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 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(-1 + \frac{4 - u \cdot u}{2 + u}\right)\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (log (+ u (* (exp (/ -2.0 v)) (+ -1.0 (/ (- 4.0 (* u u)) (+ 2.0 u)))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + (expf((-2.0f / v)) * (-1.0f + ((4.0f - (u * u)) / (2.0f + u)))))));
}

Fortran

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(exp(Float32(Float32(-2.0) / v)) * Float32(Float32(-1.0) + Float32(Float32(Float32(4.0) - Float32(u * u)) / Float32(Float32(2.0) + u))))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + (exp((single(-2.0) / v)) * (single(-1.0) + ((single(4.0) - (u * u)) / (single(2.0) + u)))))));
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. Step-by-step derivation

   1. expm1-log1p-u 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

4. Applied egg-rr 99.6%

   \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)} \cdot e^{\frac{-2}{v}}\right) \]
5. Step-by-step derivation

   1. expm1-undefine 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} - 1}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   2. sub-neg 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} + \left(-1\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   3. log1p-undefine 99.8%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

6. Simplified 99.6%

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

   1. sub-neg 99.6%

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

   2. flip-+ 99.7%

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

   3. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

4. Applied egg-rr 99.6%

   \[\leadsto 1 + v \cdot \log \left(u + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u\right)\right)} \cdot e^{\frac{-2}{v}}\right) \]
5. Step-by-step derivation

   1. expm1-undefine 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} - 1}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   2. sub-neg 99.7%

      \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(1 - u\right)} + \left(-1\right)}, e^{\frac{-2}{v}}, u\right)\right), 1\right) \]

   3. log1p-undefine 99.8%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

6. Simplified 99.6%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

Alternative 8: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + exp((single(-2.0) / 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 u around 0 97.4%

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

Alternative 9: 86.7% 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 99.6%

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

   1. +-commutative 99.6%

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

   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 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0 89.8%

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

Alternative 10: 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 99.6%

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

3. Taylor expanded in u around 0 4.8%

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

Reproduce

herbie shell --seed 2024113 
(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))))))))