HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 17.7s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + t_0 \cdot \left(1 - u\right)\right) + 0.6666666666666666 \cdot \log \left(\mathsf{fma}\left(t_0, 1 - u, u\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v))))
   (+
    1.0
    (*
     v
     (+
      (* 0.3333333333333333 (log (+ u (* t_0 (- 1.0 u)))))
      (* 0.6666666666666666 (log (fma t_0 (- 1.0 u) u))))))))

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	return 1.0f + (v * ((0.3333333333333333f * logf((u + (t_0 * (1.0f - u))))) + (0.6666666666666666f * logf(fmaf(t_0, (1.0f - u), u)))));
}

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	return Float32(Float32(1.0) + Float32(v * Float32(Float32(Float32(0.3333333333333333) * log(Float32(u + Float32(t_0 * Float32(Float32(1.0) - u))))) + Float32(Float32(0.6666666666666666) * log(fma(t_0, 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. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

   3. +-commutative 99.2%

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

   4. +-commutative 99.2%

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

   5. fma-udef 99.2%

      \[\leadsto {\left(\sqrt[3]{v \cdot \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)} + 1}\right)}^{3} \]

   6. fma-udef 99.2%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)}}\right)}^{3} \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)}\right)}^{3}} \]

5. Taylor expanded in v around 0 99.5%

   \[\leadsto \color{blue}{1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + 0.6666666666666666 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)} \]
6. Step-by-step derivation

   1. expm1-log1p-u 13.3%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)\right)}\right) \]

   2. expm1-udef 13.2%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)\right)} - 1\right)}\right) \]

   3. *-commutative 13.2%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) \cdot 0.6666666666666666}\right)} - 1\right)\right) \]

   4. +-commutative 13.2%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left(e^{\frac{-2}{v}} \cdot \left(1 - u\right) + u\right)} \cdot 0.6666666666666666\right)} - 1\right)\right) \]

   5. fma-def 13.3%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)} \cdot 0.6666666666666666\right)} - 1\right)\right) \]

7. Applied egg-rr 13.3%

   \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot 0.6666666666666666\right)} - 1\right)}\right) \]
8. Step-by-step derivation

   1. expm1-def 13.3%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot 0.6666666666666666\right)\right)}\right) \]

   2. expm1-log1p 99.6%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{\log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right) \cdot 0.6666666666666666}\right) \]

   3. *-commutative 99.6%

      \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{0.6666666666666666 \cdot \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)}\right) \]

9. Simplified 99.6%

   \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + \color{blue}{0.6666666666666666 \cdot \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)}\right) \]

10. Final simplification 99.6%

    \[\leadsto 1 + v \cdot \left(0.3333333333333333 \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + 0.6666666666666666 \cdot \log \left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)\right) \]
11. Add Preprocessing

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. 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 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 91.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ 1.0 (- (* u (* v (expm1 (/ 2.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * expm1f((2.0f / v)))) - 2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.4%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. fma-def 91.9%

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

      3. +-commutative 91.9%

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

      4. fma-def 92.3%

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

   3. Simplified 92.3%

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

      1. fma-udef 92.5%

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in u around 0 72.5%

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

      1. sub-neg 72.5%

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

      2. rec-exp 72.5%

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

      3. metadata-eval 72.5%

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

   9. Applied egg-rr 72.5%

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

       1. metadata-eval 72.5%

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

       2. sub-neg 72.5%

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

       3. distribute-neg-frac 72.5%

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

       4. metadata-eval 72.5%

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

       5. metadata-eval 72.5%

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

       6. associate-*r/ 72.5%

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

       7. expm1-def 72.5%

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

       8. associate-*r/ 72.5%

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

       9. metadata-eval 72.5%

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

   11. Simplified 72.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{--2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+ -1.0 (* u (* v (expm1 (/ (- -2.0) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (v * expm1f((-(-2.0f) / v))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * expm1(Float32(Float32(-Float32(-2.0)) / v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.4%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. fma-def 91.9%

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

      3. +-commutative 91.9%

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

      4. fma-def 92.3%

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

   3. Simplified 92.3%

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

      1. *-un-lft-identity 92.3%

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

      2. exp-prod 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. exp-1-e 92.1%

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

   8. Simplified 92.1%

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

   9. Taylor expanded in u around 0 71.6%

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

       1. associate-+r+ 71.5%

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

       2. log-E 72.5%

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

       3. metadata-eval 72.5%

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

       4. metadata-eval 72.5%

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

       5. rec-exp 72.5%

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

       6. log-E 72.5%

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

       7. associate-*r/ 72.5%

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

       8. metadata-eval 72.5%

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

       9. expm1-def 72.5%

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

   11. Simplified 72.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{--2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    1.0
    (+
     (* -2.0 (- 1.0 u))
     (* 0.5 (/ (+ (* -4.0 (* (- 1.0 u) (- 1.0 u))) (* (- 1.0 u) 4.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((-2.0f * (1.0f - u)) + (0.5f * (((-4.0f * ((1.0f - u) * (1.0f - u))) + ((1.0f - u) * 4.0f)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((-2.0e0) * (1.0e0 - u)) + (0.5e0 * ((((-4.0e0) * ((1.0e0 - u) * (1.0e0 - u))) + ((1.0e0 - u) * 4.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(-2.0) * Float32(Float32(1.0) - u)) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u))) + Float32(Float32(Float32(1.0) - u) * Float32(4.0))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(-2.0) * (single(1.0) - u)) + (single(0.5) * (((single(-4.0) * ((single(1.0) - u) * (single(1.0) - u))) + ((single(1.0) - u) * single(4.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.4%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. fma-def 91.9%

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

      3. +-commutative 91.9%

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

      4. fma-def 92.3%

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

   3. Simplified 92.3%

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

      1. fma-udef 92.5%

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in v around inf 66.4%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)} + 1 \]
   8. Step-by-step derivation

      1. unpow2 66.4%

         \[\leadsto \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot \color{blue}{\left(\left(1 - u\right) \cdot \left(1 - u\right)\right)} + 4 \cdot \left(1 - u\right)}{v}\right) + 1 \]

   9. Applied egg-rr 66.4%

      \[\leadsto \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot \color{blue}{\left(\left(1 - u\right) \cdot \left(1 - u\right)\right)} + 4 \cdot \left(1 - u\right)}{v}\right) + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{-4 \cdot \left(\left(1 - u\right) \cdot \left(1 - u\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.4%

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

   if 0.200000003 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 72.4%

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

      1. sub-neg 72.4%

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

      2. rec-exp 72.4%

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

      3. metadata-eval 72.4%

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

      4. associate-*r/ 72.4%

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

      5. metadata-eval 72.4%

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

   5. Simplified 72.4%

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

   6. Taylor expanded in v around inf 65.6%

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

      1. sub-neg 65.6%

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

      2. distribute-lft-out 65.6%

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

      3. metadata-eval 65.6%

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

   8. Simplified 65.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 5.9% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 -1.0)

C

float code(float u, float v) {
	return -1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = -1.0e0
end function

Julia

function code(u, v)
	return Float32(-1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(-1.0);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 5.4%

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

4. Final simplification 5.4%

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

Alternative 9: 86.7% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u v) :precision binary32 1.0)

C

float code(float u, float v) {
	return 1.0f;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(u, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. Taylor expanded in v around 0 88.3%

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

8. Final simplification 88.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))))