HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.0s

Alternatives: 9

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. expm1-log1p-u 94.5%

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

   2. log1p-define 94.5%

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

   3. expm1-undefine 94.5%

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

   4. add-exp-log 99.6%

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

   5. +-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-undefine 99.6%

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

   8. fma-undefine 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in v around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 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 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
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 3: 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 4: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u - (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 inf 94.9%

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

   1. neg-mul-1 94.9%

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

5. Simplified 94.9%

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

6. Final simplification 94.9%

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

Alternative 5: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(-expm1(Float32(Float32(-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. 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 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in u around inf 94.9%

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

   1. +-commutative 94.9%

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

   2. distribute-rgt-in 94.9%

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

   3. mul-1-neg 94.9%

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

   4. distribute-lft-neg-in 94.9%

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

   5. distribute-rgt-neg-out 94.9%

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

   6. *-lft-identity 94.9%

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

   7. remove-double-neg 94.9%

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

   8. neg-mul-1 94.9%

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

   9. distribute-rgt-in 94.9%

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

   10. metadata-eval 94.9%

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

   11. sub-neg 94.9%

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

   12. *-commutative 94.9%

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

   13. expm1-define 94.9%

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

7. Simplified 94.9%

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

8. Taylor expanded in v around 0 94.9%

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

   1. mul-1-neg 94.9%

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

   2. expm1-define 94.9%

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

   3. distribute-rgt-neg-in 94.9%

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

10. Simplified 94.9%

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

Alternative 6: 86.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(Float32(2.0) * Float32(u / 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 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in u around inf 94.9%

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

   1. +-commutative 94.9%

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

   2. distribute-rgt-in 94.9%

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

   3. mul-1-neg 94.9%

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

   4. distribute-lft-neg-in 94.9%

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

   5. distribute-rgt-neg-out 94.9%

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

   6. *-lft-identity 94.9%

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

   7. remove-double-neg 94.9%

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

   8. neg-mul-1 94.9%

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

   9. distribute-rgt-in 94.9%

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

   10. metadata-eval 94.9%

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

   11. sub-neg 94.9%

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

   12. *-commutative 94.9%

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

   13. expm1-define 94.9%

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

7. Simplified 94.9%

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

8. Taylor expanded in v around inf 87.4%

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

Alternative 7: 86.6% accurate, 2.0× speedup

Math

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

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(u * Float32(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 99.6%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in u around inf 94.9%

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

   1. +-commutative 94.9%

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

   2. distribute-rgt-in 94.9%

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

   3. mul-1-neg 94.9%

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

   4. distribute-lft-neg-in 94.9%

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

   5. distribute-rgt-neg-out 94.9%

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

   6. *-lft-identity 94.9%

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

   7. remove-double-neg 94.9%

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

   8. neg-mul-1 94.9%

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

   9. distribute-rgt-in 94.9%

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

   10. metadata-eval 94.9%

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

   11. sub-neg 94.9%

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

   12. *-commutative 94.9%

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

   13. expm1-define 94.9%

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

7. Simplified 94.9%

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

8. Taylor expanded in v around 0 94.9%

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

   1. mul-1-neg 94.9%

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

   2. expm1-define 94.9%

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

   3. distribute-rgt-neg-in 94.9%

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

10. Simplified 94.9%

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

11. Taylor expanded in v around inf 87.4%

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

Alternative 8: 86.3% 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 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
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 87.2%

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

Alternative 9: 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 99.6%

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

3. Taylor expanded in u around 0 5.3%

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

Reproduce

herbie shell --seed 2024123 
(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))))))))