HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 21.8s

Alternatives: 18

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: 99.4% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\\ \mathsf{fma}\left(v, \frac{-1 + {\left(1 + t_0\right)}^{2}}{t_0 + 2}, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (log (fma (- 1.0 u) (exp (/ -2.0 v)) u))))
   (fma v (/ (+ -1.0 (pow (+ 1.0 t_0) 2.0)) (+ t_0 2.0)) 1.0)))

C

float code(float u, float v) {
	float t_0 = logf(fmaf((1.0f - u), expf((-2.0f / v)), u));
	return fmaf(v, ((-1.0f + powf((1.0f + t_0), 2.0f)) / (t_0 + 2.0f)), 1.0f);
}

Julia

function code(u, v)
	t_0 = log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u))
	return fma(v, Float32(Float32(Float32(-1.0) + (Float32(Float32(1.0) + t_0) ^ Float32(2.0))) / Float32(t_0 + Float32(2.0))), Float32(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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

   1. fma-def 99.5%

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

   2. expm1-log1p-u 13.0%

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

   3. expm1-def 13.0%

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

   4. flip-- 13.0%

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

   5. div-inv 13.0%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r+ 99.6%

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

   5. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

   1. fma-def 99.5%

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

   2. expm1-log1p-u 13.0%

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

   3. expm1-def 13.0%

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

   4. log1p-udef 13.1%

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

   5. rem-exp-log 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(fma(Float32(Float32(1.0) - u), (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)), u)), Float32(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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

6. Applied egg-rr 99.6%

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

   1. exp-1-e 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

4. Applied egg-rr 99.5%

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

   1. exp-1-e 99.6%

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

6. Simplified 99.5%

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

   1. add-log-exp 99.4%

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

   2. *-commutative 99.4%

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

   3. exp-to-pow 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-def 99.5%

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

   6. e-exp-1 99.5%

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

   7. pow-exp 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. add-exp-log 99.5%

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

   10. exp-prod 99.5%

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

   11. *-commutative 99.5%

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

   12. add-log-exp 99.5%

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

   13. expm1-log1p-u 13.0%

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

   14. expm1-def 13.0%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(v, log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v))))), Float32(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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

8. Applied egg-rr 99.5%

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

   1. exp-1-e 99.6%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 6: 99.4% 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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(2.71828182845904523536) ^ (single(-2.0) / v))))));
end

Derivation

1. Initial program 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

4. Applied egg-rr 99.5%

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

   1. exp-1-e 99.6%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 10: 90.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \left(-1 + e^{\frac{2}{v}}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ 1.0 (- (* u (* v (+ -1.0 (exp (/ 2.0 v))))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * (-1.0f + expf((2.0f / v))))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (v * ((-1.0e0) + exp((2.0e0 / v))))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * Float32(Float32(-1.0) + exp(Float32(Float32(2.0) / v))))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (v * (single(-1.0) + exp((single(2.0) / v))))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0 70.0%

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

      1. sub-neg 70.0%

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

      2. rec-exp 70.0%

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

      3. metadata-eval 70.0%

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

      4. associate-*r/ 70.0%

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

      5. metadata-eval 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in v around 0 70.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \left(-1 + e^{\frac{2}{v}}\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + \mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ 1.0 (+ -2.0 (* (expm1 (/ 2.0 v)) (* v u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f + (expm1f((2.0f / v)) * (v * u)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) + Float32(expm1(Float32(Float32(2.0) / v)) * Float32(v * u))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

      1. add-log-exp 94.3%

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

      2. *-commutative 94.3%

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

      3. exp-to-pow 94.5%

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

      4. +-commutative 94.5%

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

      5. fma-udef 94.8%

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

   4. Applied egg-rr 94.8%

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

      1. *-un-lft-identity 94.6%

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

      2. exp-prod 94.9%

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

   6. Applied egg-rr 95.1%

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

      1. exp-1-e 94.9%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in u around 0 69.0%

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

       1. log-pow 69.0%

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

       2. log-E 70.2%

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

       3. metadata-eval 70.2%

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

       4. associate-*r* 70.2%

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

       5. rec-exp 70.2%

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

       6. log-E 70.2%

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

       7. associate-*r/ 70.2%

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

       8. metadata-eval 70.2%

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

       9. expm1-def 70.2%

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

   11. Simplified 70.2%

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

   12. Taylor expanded in u around 0 70.2%

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

       1. associate-*r* 70.2%

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

       2. *-commutative 70.2%

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

       3. expm1-def 70.2%

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

   14. Simplified 70.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + \mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{--2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ -1.0 (* (* v u) (expm1 (/ (- -2.0) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((v * u) * expm1f((-(-2.0f) / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(v * u) * expm1(Float32(Float32(-Float32(-2.0)) / v))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. fma-def 94.4%

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

      3. +-commutative 94.4%

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

      4. fma-def 94.6%

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

   3. Simplified 94.6%

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

      1. *-un-lft-identity 94.6%

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

      2. exp-prod 94.9%

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

   6. Applied egg-rr 94.9%

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

      1. exp-1-e 94.9%

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

   8. Simplified 94.9%

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

   9. Taylor expanded in u around 0 69.0%

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

       1. associate-+r+ 68.8%

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

       2. log-E 70.2%

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

       3. metadata-eval 70.2%

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

       4. metadata-eval 70.2%

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

       5. associate-*r* 70.2%

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

       6. rec-exp 70.2%

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

       7. log-E 70.2%

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

       8. associate-*r/ 70.2%

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

       9. metadata-eval 70.2%

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

       10. expm1-def 70.2%

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

   11. Simplified 70.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(v \cdot u\right) \cdot \mathsf{expm1}\left(\frac{--2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.1% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + 2 \cdot \left(u + \frac{u}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ 1.0 (+ -2.0 (* 2.0 (+ u (/ u v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (-2.0f + (2.0f * (u + (u / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((-2.0e0) + (2.0e0 * (u + (u / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(-2.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (single(-2.0) + (single(2.0) * (u + (u / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0 70.0%

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

      1. sub-neg 70.0%

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

      2. rec-exp 70.0%

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

      3. metadata-eval 70.0%

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

      4. associate-*r/ 70.0%

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

      5. metadata-eval 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in v around inf 57.6%

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

      1. sub-neg 57.6%

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

      2. distribute-lft-out 57.6%

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

      3. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 + 2 \cdot \left(u + \frac{u}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (2.0f * (u + (u / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (2.0e0 * (u + (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(2.0) * Float32(u + Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (single(2.0) * (u + (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in u around 0 70.0%

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

      1. sub-neg 70.0%

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

      2. rec-exp 70.0%

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

      3. metadata-eval 70.0%

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

      4. associate-*r/ 70.0%

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

      5. metadata-eval 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in v around inf 57.6%

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

      1. sub-neg 57.6%

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

      2. distribute-lft-out 57.6%

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

      3. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.4% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((1.0f - u) * -2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in v around inf 51.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 89.4% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.4%

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

   if 0.150000006 < v

   1. Initial program 94.1%

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

   3. Taylor expanded in v around inf 51.6%

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

   4. Taylor expanded in u around 0 51.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 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. Add Preprocessing

3. Taylor expanded in u around 0 6.2%

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

4. Final simplification 6.2%

   \[\leadsto -1 \]
5. Add Preprocessing

Alternative 18: 86.1% 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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0 85.1%

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

8. Final simplification 85.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))))))