HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 21.0s

Alternatives: 12

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. flip-+ 99.5%

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

   2. clear-num 99.5%

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

   3. log-div 99.6%

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

   4. metadata-eval 99.6%

      \[\leadsto 1 + v \cdot \left(\color{blue}{0} - \log \left(\frac{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}{u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right) \]

   5. pow2 99.6%

      \[\leadsto 1 + v \cdot \left(0 - \log \left(\frac{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}{u \cdot u - \color{blue}{{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2}}}\right)\right) \]

3. Applied egg-rr 99.6%

   \[\leadsto 1 + v \cdot \color{blue}{\left(0 - \log \left(\frac{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}{u \cdot u - {\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2}}\right)\right)} \]

4. Taylor expanded in v around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. log-div 93.7%

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

   4. *-commutative 93.7%

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

   5. *-lft-identity 93.7%

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

   6. unpow2 93.7%

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

   7. *-commutative 93.7%

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

   8. unpow2 93.7%

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

   9. unpow2 93.7%

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

6. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. clear-num 99.6%

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

   3. unpow2 99.6%

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

   4. flip-+ 99.5%

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

8. Applied egg-rr 99.5%

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

   1. *-lft-identity 99.5%

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

   2. +-commutative 99.5%

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

   3. fma-def 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in v around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-udef 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. flip-+ 99.5%

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

   2. clear-num 99.5%

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

   3. log-div 99.6%

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

   4. metadata-eval 99.6%

      \[\leadsto 1 + v \cdot \left(\color{blue}{0} - \log \left(\frac{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}{u \cdot u - \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}\right)\right) \]

   5. pow2 99.6%

      \[\leadsto 1 + v \cdot \left(0 - \log \left(\frac{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}{u \cdot u - \color{blue}{{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2}}}\right)\right) \]

3. Applied egg-rr 99.6%

   \[\leadsto 1 + v \cdot \color{blue}{\left(0 - \log \left(\frac{u - \left(1 - u\right) \cdot e^{\frac{-2}{v}}}{u \cdot u - {\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}^{2}}\right)\right)} \]

4. Taylor expanded in v around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. log-div 93.7%

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

   4. *-commutative 93.7%

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

   5. *-lft-identity 93.7%

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

   6. unpow2 93.7%

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

   7. *-commutative 93.7%

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

   8. unpow2 93.7%

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

   9. unpow2 93.7%

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

6. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. clear-num 99.6%

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

   3. unpow2 99.6%

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

   4. flip-+ 99.5%

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

8. Applied egg-rr 99.5%

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

   1. *-lft-identity 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 4: 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 99.5%

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

2. Final simplification 99.5%

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

Alternative 5: 90.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ (* u (* v (+ (/ 1.0 (exp (/ -2.0 v))) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * ((1.0f / expf((-2.0f / v))) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * ((1.0e0 / exp(((-2.0e0) / v))) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32(Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (u * (v * ((single(1.0) / exp((single(-2.0) / v))) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in v around 0 94.7%

      \[\leadsto \color{blue}{1} \]

   if 0.100000001 < v

   1. Initial program 93.2%

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

   2. Taylor expanded in u around 0 67.5%

      \[\leadsto \color{blue}{u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} + -1\right)\right) + -1\\ \end{array} \]

Alternative 6: 90.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ (* (* v u) (expm1 (/ 2.0 v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = ((v * u) * expm1f((2.0f / v))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(v * u) * expm1(Float32(Float32(2.0) / v))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in v around 0 94.7%

      \[\leadsto \color{blue}{1} \]

   if 0.100000001 < v

   1. Initial program 93.2%

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

   2. Taylor expanded in u around 0 67.2%

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

      1. associate-+r- 67.2%

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

      2. rec-exp 67.2%

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

      3. expm1-def 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. associate--l+ 67.2%

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

      2. associate-*r* 67.2%

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

      3. distribute-neg-frac 67.2%

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

      4. metadata-eval 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in u around 0 67.3%

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

      1. expm1-log1p-u 67.3%

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

      2. expm1-udef 67.2%

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

      3. expm1-def 67.2%

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

   9. Applied egg-rr 67.2%

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

       1. expm1-def 67.5%

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

       2. expm1-log1p 67.5%

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

       3. associate-*r* 67.5%

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

   11. Simplified 67.5%

       \[\leadsto \color{blue}{\left(u \cdot v\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + -1\\ \end{array} \]

Alternative 7: 90.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ (* u (+ (/ 2.0 v) (+ 2.0 (/ 1.3333333333333333 (* v v))))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (u * ((2.0f / v) + (2.0f + (1.3333333333333333f / (v * v))))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (u * ((2.0e0 / v) + (2.0e0 + (1.3333333333333333e0 / (v * v))))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(Float32(2.0) / v) + Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (u * ((single(2.0) / v) + (single(2.0) + (single(1.3333333333333333) / (v * v))))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in v around 0 94.7%

      \[\leadsto \color{blue}{1} \]

   if 0.100000001 < v

   1. Initial program 93.2%

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

   2. Taylor expanded in u around 0 67.2%

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

      1. associate-+r- 67.2%

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

      2. rec-exp 67.2%

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

      3. expm1-def 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. associate--l+ 67.2%

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

      2. associate-*r* 67.2%

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

      3. distribute-neg-frac 67.2%

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

      4. metadata-eval 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in u around 0 67.3%

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

   8. Taylor expanded in v around inf 63.2%

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

      1. associate-+r+ 63.2%

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

      2. associate-*r/ 63.2%

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

      3. metadata-eval 63.2%

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

      4. unpow2 63.2%

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

      5. associate-*r/ 63.2%

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

      6. metadata-eval 63.2%

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

   10. Simplified 63.2%

       \[\leadsto u \cdot \color{blue}{\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \frac{2}{v}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\frac{2}{v} + \left(2 + \frac{1.3333333333333333}{v \cdot v}\right)\right) + -1\\ \end{array} \]

Alternative 8: 90.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612) 1.0 (+ (* 2.0 (+ u (/ u v))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * (u + (u / v))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * (u + (u / v))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(u + Float32(u / v))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * (u + (u / v))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in v around 0 94.7%

      \[\leadsto \color{blue}{1} \]

   if 0.100000001 < v

   1. Initial program 93.2%

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

   2. Taylor expanded in u around 0 67.2%

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

   3. Taylor expanded in v around inf 60.2%

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

      1. sub-neg 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. metadata-eval 60.2%

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

   5. Simplified 60.2%

      \[\leadsto \color{blue}{2 \cdot \left(u + \frac{u}{v}\right) + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \]

Alternative 9: 89.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25999999046325684:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25999999046325684) 1.0 (+ 1.0 (* -2.0 (- 1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25999999046325684f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f * (1.0f - u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25999999046325684e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) * (1.0e0 - u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25999999046325684))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(1.0) - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25999999046325684))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) * (single(1.0) - u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25999999

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in v around 0 93.7%

      \[\leadsto \color{blue}{1} \]

   if 0.25999999 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in v around inf 56.9%

      \[\leadsto 1 + \color{blue}{-2 \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25999999046325684:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 10: 89.5% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25999999046325684:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25999999046325684) 1.0 (+ (* u 2.0) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25999999046325684f) {
		tmp = 1.0f;
	} else {
		tmp = (u * 2.0f) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25999999046325684e0) then
        tmp = 1.0e0
    else
        tmp = (u * 2.0e0) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25999999046325684))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(2.0)) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25999999046325684))
		tmp = single(1.0);
	else
		tmp = (u * single(2.0)) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25999999

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in v around 0 93.7%

      \[\leadsto \color{blue}{1} \]

   if 0.25999999 < v

   1. Initial program 92.5%

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

   2. Taylor expanded in v around inf 56.9%

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

   3. Taylor expanded in u around 0 56.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25999999046325684:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \]

Alternative 11: 6.1% 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 99.5%

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

2. Taylor expanded in u around 0 6.0%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 6.0%

   \[\leadsto -1 \]

Alternative 12: 86.2% 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 99.5%

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

2. Taylor expanded in v around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-udef 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in v around 0 88.1%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 88.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(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))))))))