HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.6% → 99.5%

Time: 24.9s

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. expm1-log1p-u 12.8%

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

   2. expm1-udef 12.9%

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

   3. +-commutative 12.9%

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

   4. fma-udef 12.9%

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

3. Applied egg-rr 12.9%

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

   1. sub-neg 12.9%

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

   2. log1p-udef 12.9%

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

   3. add-exp-log 99.6%

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

   4. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. expm1-log1p-u 12.8%

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

   2. expm1-udef 12.9%

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

   3. +-commutative 12.9%

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

   4. fma-udef 12.9%

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

3. Applied egg-rr 12.9%

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

   1. sub-neg 12.9%

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

   2. log1p-udef 12.9%

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

   3. add-exp-log 99.6%

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

   4. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Final simplification 99.5%

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

Alternative 4: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(-expm1(Float32(Float32(-2.0) / v)))))))
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. Taylor expanded in u around -inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. neg-mul-1 99.5%

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

   3. expm1-def 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in u around inf 94.9%

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

   1. +-commutative 94.9%

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

   2. mul-1-neg 94.9%

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

   3. log-rec 94.9%

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

   4. remove-double-neg 94.9%

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

   5. log-prod 94.9%

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

   6. mul-1-neg 94.9%

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

   7. expm1-def 94.9%

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

9. Simplified 94.9%

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

10. Final simplification 94.9%

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

Alternative 5: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + 0.5 \cdot \frac{\left(4 + \left(u \cdot -2\right) \cdot 2\right) - {\left(u \cdot -2 + 2\right)}^{2}}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+
    (- -1.0 (* u -2.0))
    (*
     0.5
     (/ (- (+ 4.0 (* (* u -2.0) 2.0)) (pow (+ (* u -2.0) 2.0) 2.0)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = (-1.0f - (u * -2.0f)) + (0.5f * (((4.0f + ((u * -2.0f) * 2.0f)) - powf(((u * -2.0f) + 2.0f), 2.0f)) / 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 <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = ((-1.0e0) - (u * (-2.0e0))) + (0.5e0 * (((4.0e0 + ((u * (-2.0e0)) * 2.0e0)) - (((u * (-2.0e0)) + 2.0e0) ** 2.0e0)) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(-1.0) - Float32(u * Float32(-2.0))) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(4.0) + Float32(Float32(u * Float32(-2.0)) * Float32(2.0))) - (Float32(Float32(u * Float32(-2.0)) + Float32(2.0)) ^ Float32(2.0))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.1599999964237213))
		tmp = single(1.0);
	else
		tmp = (single(-1.0) - (u * single(-2.0))) + (single(0.5) * (((single(4.0) + ((u * single(-2.0)) * single(2.0))) - (((u * single(-2.0)) + single(2.0)) ^ single(2.0))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   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. Taylor expanded in u around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. expm1-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.159999996 < v

   1. Initial program 91.9%

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

      1. +-commutative 91.9%

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

      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 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in u around -inf 93.0%

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

      1. associate-*r* 93.0%

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

      2. neg-mul-1 93.0%

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

      3. expm1-def 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   8. Taylor expanded in v around -inf 57.8%

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

      1. associate-+r+ 57.7%

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

      2. distribute-lft-in 57.7%

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

      3. metadata-eval 57.7%

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

      4. associate-+r+ 57.9%

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

      5. metadata-eval 57.9%

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

      6. mul-1-neg 57.9%

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

      7. *-commutative 57.9%

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

   10. Simplified 57.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - u \cdot -2\right) + 0.5 \cdot \frac{\left(4 + \left(u \cdot -2\right) \cdot 2\right) - {\left(u \cdot -2 + 2\right)}^{2}}{v}\\ \end{array} \]

Alternative 6: 91.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{\mathsf{fma}\left(-2, u, 2 \cdot \left(u \cdot u\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+ -1.0 (- (* u 2.0) (/ (fma -2.0 u (* 2.0 (* u u))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (fmaf(-2.0f, u, (2.0f * (u * u))) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(fma(Float32(-2.0), u, Float32(Float32(2.0) * Float32(u * u))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   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. Taylor expanded in u around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. expm1-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.159999996 < v

   1. Initial program 91.9%

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

   2. Taylor expanded in v around inf 57.8%

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

   3. Taylor expanded in u around 0 57.8%

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

   4. Taylor expanded in v around -inf 57.8%

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

      1. +-commutative 57.8%

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

      2. mul-1-neg 57.8%

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

      3. unsub-neg 57.8%

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

      4. *-commutative 57.8%

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

      5. fma-def 57.8%

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

      6. unpow2 57.8%

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

   6. Simplified 57.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{\mathsf{fma}\left(-2, u, 2 \cdot \left(u \cdot u\right)\right)}{v}\right)\\ \end{array} \]

Alternative 7: 91.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (* 0.5 (/ (+ (* -4.0 (+ 1.0 (* u (+ u -2.0)))) (* (- 1.0 u) 4.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (0.5f * (((-4.0f * (1.0f + (u * (u + -2.0f)))) + ((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 <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (0.5e0 * ((((-4.0e0) * (1.0e0 + (u * (u + (-2.0e0))))) + ((1.0e0 - u) * 4.0e0)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(1.0) + Float32(u * Float32(u + Float32(-2.0))))) + 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(0.1599999964237213))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (single(0.5) * (((single(-4.0) * (single(1.0) + (u * (u + single(-2.0))))) + ((single(1.0) - u) * single(4.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   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. Taylor expanded in u around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. expm1-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 92.8%

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

   if 0.159999996 < v

   1. Initial program 91.9%

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

   2. Taylor expanded in v around inf 57.8%

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

   3. Taylor expanded in u around 0 57.8%

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

      1. unpow2 57.8%

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

      2. distribute-rgt-out 57.8%

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

   5. Simplified 57.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{-4 \cdot \left(1 + u \cdot \left(u + -2\right)\right) + \left(1 - u\right) \cdot 4}{v}\right)\\ \end{array} \]

Alternative 8: 5.7% 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. Taylor expanded in u around 0 5.3%

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

3. Final simplification 5.3%

   \[\leadsto -1 \]

Alternative 9: 87.4% 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. Taylor expanded in u around -inf 99.5%

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

   1. associate-*r* 99.5%

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

   2. neg-mul-1 99.5%

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

   3. expm1-def 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in v around 0 87.2%

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

8. Final simplification 87.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023287 
(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))))))))