HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.3s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(\mathsf{fma}\left(\frac{1 - {u}^{3}}{1 + \left(u + u \cdot u\right)}, e^{\frac{-2}{v}}, u\right)\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (+
  1.0
  (*
   v
   (log
    (fma (/ (- 1.0 (pow u 3.0)) (+ 1.0 (+ u (* u u)))) (exp (/ -2.0 v)) u)))))

C

float code(float u, float v) {
	return 1.0f + (v * logf(fmaf(((1.0f - powf(u, 3.0f)) / (1.0f + (u + (u * u)))), expf((-2.0f / v)), u)));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(fma(Float32(Float32(Float32(1.0) - (u ^ Float32(3.0))) / Float32(Float32(1.0) + Float32(u + Float32(u * u)))), exp(Float32(Float32(-2.0) / v)), u))))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. fma-define 99.6%

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

5. Simplified 99.6%

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

   1. sub-neg 99.6%

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

   2. flip3-+ 99.6%

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

   3. metadata-eval 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-un-lft-identity 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in u around 0 99.6%

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

   1. neg-mul-1 99.6%

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

   2. sub-neg 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

    \[\leadsto 1 + v \cdot \log \left(\mathsf{fma}\left(\frac{1 - {u}^{3}}{1 + \left(u + u \cdot u\right)}, e^{\frac{-2}{v}}, u\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 3: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0 95.4%

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

Alternative 4: 95.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log u \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (+ 1.0 (* v (log u))))

C

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

Fortran

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(u)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0 95.4%

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

4. Taylor expanded in u around inf 94.3%

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

   1. mul-1-neg 94.3%

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

   2. distribute-rgt-neg-in 94.3%

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

   3. log-rec 94.3%

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

   4. remove-double-neg 94.3%

      \[\leadsto 1 + v \cdot \color{blue}{\log u} \]

6. Simplified 94.3%

   \[\leadsto 1 + \color{blue}{v \cdot \log u} \]
7. Add Preprocessing

Alternative 5: 90.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   1.0
   (+
    -1.0
    (+
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (* u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) + (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 <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) + (u * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) + Float32(u * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.2199999988079071))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) + (u * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   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-define 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-define 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. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. *-commutative 97.1%

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

      7. exp-to-pow 97.1%

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

      8. +-commutative 97.1%

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

      9. fma-undefine 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in v around 0 93.2%

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

   if 0.219999999 < v

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. fma-define 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-define 94.0%

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

   3. Simplified 94.0%

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

      1. fma-undefine 94.0%

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

      2. fma-undefine 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. expm1-log1p-u 94.3%

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

      6. expm1-undefine 94.3%

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

      7. +-commutative 94.3%

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

      8. +-commutative 94.3%

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

      9. fma-undefine 93.9%

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

      10. fma-undefine 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in u around 0 56.9%

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

      1. associate-*r* 57.0%

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

      2. sub-neg 57.0%

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

      3. rec-exp 57.0%

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

      4. metadata-eval 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in v around -inf 57.6%

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{-2 \cdot u + -1 \cdot \frac{0.6666666666666666 \cdot \frac{u}{v} + 1.3333333333333333 \cdot u}{v}}{v} + 2 \cdot u\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} + u \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 - u \cdot \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   1.0
   (+
    -1.0
    (-
     (* u 2.0)
     (/
      (-
       (* u -2.0)
       (* u (/ (+ 1.3333333333333333 (/ 0.6666666666666666 v)) v)))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((u * -2.0f) - (u * ((1.3333333333333333f + (0.6666666666666666f / v)) / v))) / 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 <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - (((u * (-2.0e0)) - (u * ((1.3333333333333333e0 + (0.6666666666666666e0 / v)) / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(-2.0)) - Float32(u * Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) / v)) / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.2199999988079071))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (((u * single(-2.0)) - (u * ((single(1.3333333333333333) + (single(0.6666666666666666) / v)) / v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   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-define 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-define 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. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. *-commutative 97.1%

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

      7. exp-to-pow 97.1%

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

      8. +-commutative 97.1%

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

      9. fma-undefine 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in v around 0 93.2%

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

   if 0.219999999 < v

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. fma-define 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-define 94.0%

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

   3. Simplified 94.0%

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

      1. fma-undefine 94.0%

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

      2. fma-undefine 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. expm1-log1p-u 94.3%

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

      6. expm1-undefine 94.3%

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

      7. +-commutative 94.3%

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

      8. +-commutative 94.3%

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

      9. fma-undefine 93.9%

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

      10. fma-undefine 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in u around 0 56.9%

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

      1. associate-*r* 57.0%

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

      2. sub-neg 57.0%

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

      3. rec-exp 57.0%

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

      4. metadata-eval 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in v around -inf 57.6%

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{-2 \cdot u + -1 \cdot \frac{0.6666666666666666 \cdot \frac{u}{v} + 1.3333333333333333 \cdot u}{v}}{v} + 2 \cdot u\right)} - 1 \]

   11. Taylor expanded in u around 0 57.6%

       \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \color{blue}{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}\right)}{v}}}{v} + 2 \cdot u\right) - 1 \]
   12. Step-by-step derivation

       1. mul-1-neg 57.6%

          \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \color{blue}{\left(-\frac{u \cdot \left(1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}\right)}{v}\right)}}{v} + 2 \cdot u\right) - 1 \]

       2. associate-/l* 57.6%

          \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \left(-\color{blue}{u \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}\right)}{v} + 2 \cdot u\right) - 1 \]

       3. distribute-lft-neg-in 57.6%

          \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \color{blue}{\left(-u\right) \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}}{v} + 2 \cdot u\right) - 1 \]

       4. associate-*r/ 57.6%

          \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \left(-u\right) \cdot \frac{1.3333333333333333 + \color{blue}{\frac{0.6666666666666666 \cdot 1}{v}}}{v}}{v} + 2 \cdot u\right) - 1 \]

       5. metadata-eval 57.6%

          \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \left(-u\right) \cdot \frac{1.3333333333333333 + \frac{\color{blue}{0.6666666666666666}}{v}}{v}}{v} + 2 \cdot u\right) - 1 \]

   13. Simplified 57.6%

       \[\leadsto \left(-1 \cdot \frac{-2 \cdot u + \color{blue}{\left(-u\right) \cdot \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}}}{v} + 2 \cdot u\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot -2 - u \cdot \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-2 \cdot \left(1 - u\right) + 0.5 \cdot \frac{u \cdot \left(4 + u \cdot -4\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   1.0
   (+ 1.0 (+ (* -2.0 (- 1.0 u)) (* 0.5 (/ (* u (+ 4.0 (* u -4.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((-2.0f * (1.0f - u)) + (0.5f * ((u * (4.0f + (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 <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((-2.0e0) * (1.0e0 - u)) + (0.5e0 * ((u * (4.0e0 + (u * (-4.0e0)))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		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(u * Float32(Float32(4.0) + Float32(u * Float32(-4.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.2199999988079071))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(-2.0) * (single(1.0) - u)) + (single(0.5) * ((u * (single(4.0) + (u * single(-4.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   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-define 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-define 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. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. *-commutative 97.1%

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

      7. exp-to-pow 97.1%

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

      8. +-commutative 97.1%

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

      9. fma-undefine 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in v around 0 93.2%

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

   if 0.219999999 < v

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. fma-define 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-define 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in v around inf 54.4%

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

   6. Taylor expanded in u around 0 54.4%

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

      1. *-commutative 54.4%

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

   8. Simplified 54.4%

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

Alternative 8: 90.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} + 2 \cdot \frac{1}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071)
   1.0
   (+ -1.0 (* u (+ 2.0 (+ (* -2.0 (/ u v)) (* 2.0 (/ 1.0 v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((-2.0f * (u / v)) + (2.0f * (1.0f / 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 <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (((-2.0e0) * (u / v)) + (2.0e0 * (1.0e0 / v)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.2199999988079071))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(-2.0) * Float32(u / v)) + Float32(Float32(2.0) * Float32(Float32(1.0) / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.2199999988079071))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(-2.0) * (u / v)) + (single(2.0) * (single(1.0) / v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   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-define 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-define 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. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. *-commutative 97.1%

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

      7. exp-to-pow 97.1%

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

      8. +-commutative 97.1%

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

      9. fma-undefine 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in v around 0 93.2%

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

   if 0.219999999 < v

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. fma-define 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-define 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in v around inf 54.4%

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

   6. Taylor expanded in u around 0 54.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} + 2 \cdot \frac{1}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.2199999988079071) 1.0 (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		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 <= 0.2199999988079071e0) 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(0.2199999988079071))
		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(0.2199999988079071))
		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 < 0.219999999

   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-define 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-define 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. add-log-exp 97.2%

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

      2. fma-undefine 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. exp-sum 97.1%

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

      6. *-commutative 97.1%

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

      7. exp-to-pow 97.1%

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

      8. +-commutative 97.1%

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

      9. fma-undefine 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in v around 0 93.2%

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

   if 0.219999999 < v

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. fma-define 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-define 94.0%

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

   3. Simplified 94.0%

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

      1. fma-undefine 94.0%

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

      2. fma-undefine 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. expm1-log1p-u 94.3%

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

      6. expm1-undefine 94.3%

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

      7. +-commutative 94.3%

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

      8. +-commutative 94.3%

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

      9. fma-undefine 93.9%

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

      10. fma-undefine 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in u around 0 56.9%

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

      1. associate-*r* 57.0%

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

      2. sub-neg 57.0%

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

      3. rec-exp 57.0%

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

      4. metadata-eval 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in v around inf 50.7%

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

       1. distribute-lft-out 50.7%

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

   12. Simplified 50.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.2199999988079071:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.2% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. add-log-exp 97.0%

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

   2. fma-undefine 97.0%

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

   3. fma-undefine 97.0%

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

   4. +-commutative 97.0%

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

   5. exp-sum 96.8%

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

   6. *-commutative 96.8%

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

   7. exp-to-pow 96.8%

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

   8. +-commutative 96.8%

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

   9. fma-undefine 96.8%

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

6. Applied egg-rr 96.8%

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

7. Taylor expanded in v around 0 86.9%

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

Alternative 11: 5.5% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in u around 0 5.4%

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

Reproduce

herbie shell --seed 2024141 
(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))))))))