HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.0s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% 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

Derivation

1. Initial program 99.2%

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

Alternative 2: 96.2% 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 99.2%

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

3. Taylor expanded in u around 0 95.7%

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

Alternative 3: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + u \cdot \left(\frac{1}{v} \cdot 1.3333333333333333 + u \cdot \left(4 \cdot \frac{-1}{v} - -2.6666666666666665 \cdot \frac{u}{v}\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (+
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (pow (- 1.0 u) 2.0))))
       (*
        u
        (+
         (* (/ 1.0 v) 1.3333333333333333)
         (* u (- (* 4.0 (/ -1.0 v)) (* -2.6666666666666665 (/ u v)))))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * powf((1.0f - u), 2.0f)))) + (u * (((1.0f / v) * 1.3333333333333333f) + (u * ((4.0f * (-1.0f / v)) - (-2.6666666666666665f * (u / 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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * ((1.0e0 - u) ** 2.0e0)))) + (u * (((1.0e0 / v) * 1.3333333333333333e0) + (u * ((4.0e0 * ((-1.0e0) / v)) - ((-2.6666666666666665e0) * (u / v))))))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))))) + Float32(u * Float32(Float32(Float32(Float32(1.0) / v) * Float32(1.3333333333333333)) + Float32(u * Float32(Float32(Float32(4.0) * Float32(Float32(-1.0) / v)) - Float32(Float32(-2.6666666666666665) * Float32(u / v))))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * ((single(1.0) - u) ^ single(2.0))))) + (u * (((single(1.0) / v) * single(1.3333333333333333)) + (u * ((single(4.0) * (single(-1.0) / v)) - (single(-2.6666666666666665) * (u / v))))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in v around -inf 78.5%

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0 78.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \color{blue}{u \cdot \left(u \cdot \left(-2.6666666666666665 \cdot \frac{u}{v} + 4 \cdot \frac{1}{v}\right) - 1.3333333333333333 \cdot \frac{1}{v}\right)}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + u \cdot \left(\frac{1}{v} \cdot 1.3333333333333333 + u \cdot \left(4 \cdot \frac{-1}{v} - -2.6666666666666665 \cdot \frac{u}{v}\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + u \cdot \left(\frac{1.3333333333333333}{v} - \frac{u \cdot 4}{v}\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (+
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (pow (- 1.0 u) 2.0))))
       (* u (- (/ 1.3333333333333333 v) (/ (* u 4.0) v))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * powf((1.0f - u), 2.0f)))) + (u * ((1.3333333333333333f / v) - ((u * 4.0f) / 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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * ((1.0e0 - u) ** 2.0e0)))) + (u * ((1.3333333333333333e0 / v) - ((u * 4.0e0) / v)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))))) + Float32(u * Float32(Float32(Float32(1.3333333333333333) / v) - Float32(Float32(u * Float32(4.0)) / v)))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * ((single(1.0) - u) ^ single(2.0))))) + (u * ((single(1.3333333333333333) / v) - ((u * single(4.0)) / v)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in v around -inf 78.5%

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0 76.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \color{blue}{u \cdot \left(4 \cdot \frac{u}{v} - 1.3333333333333333 \cdot \frac{1}{v}\right)}}{v}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 76.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + u \cdot \left(\color{blue}{\frac{4 \cdot u}{v}} - 1.3333333333333333 \cdot \frac{1}{v}\right)}{v}\right) \]

      2. *-commutative 76.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + u \cdot \left(\frac{\color{blue}{u \cdot 4}}{v} - 1.3333333333333333 \cdot \frac{1}{v}\right)}{v}\right) \]

      3. associate-*r/ 76.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + u \cdot \left(\frac{u \cdot 4}{v} - \color{blue}{\frac{1.3333333333333333 \cdot 1}{v}}\right)}{v}\right) \]

      4. metadata-eval 76.5%

         \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + u \cdot \left(\frac{u \cdot 4}{v} - \frac{\color{blue}{1.3333333333333333}}{v}\right)}{v}\right) \]

   8. Simplified 76.5%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \color{blue}{u \cdot \left(\frac{u \cdot 4}{v} - \frac{1.3333333333333333}{v}\right)}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) + u \cdot \left(\frac{1.3333333333333333}{v} - \frac{u \cdot 4}{v}\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) - \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (/
      (-
       (* -0.5 (- (* 4.0 (+ u -1.0)) (* -4.0 (pow (- 1.0 u) 2.0))))
       (* (/ u v) -1.3333333333333333))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (((-0.5f * ((4.0f * (u + -1.0f)) - (-4.0f * powf((1.0f - u), 2.0f)))) - ((u / v) * -1.3333333333333333f)) / 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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((-0.5e0) * ((4.0e0 * (u + (-1.0e0))) - ((-4.0e0) * ((1.0e0 - u) ** 2.0e0)))) - ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(4.0) * Float32(u + Float32(-1.0))) - Float32(Float32(-4.0) * (Float32(Float32(1.0) - u) ^ Float32(2.0))))) - Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (((single(-0.5) * ((single(4.0) * (u + single(-1.0))) - (single(-4.0) * ((single(1.0) - u) ^ single(2.0))))) - ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in v around -inf 78.5%

      \[\leadsto \color{blue}{1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + 0.16666666666666666 \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   6. Taylor expanded in u around 0 72.9%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{-0.5 \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \color{blue}{-1.3333333333333333 \cdot \frac{u}{v}}}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \frac{-0.5 \cdot \left(4 \cdot \left(u + -1\right) - -4 \cdot {\left(1 - u\right)}^{2}\right) - \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} - 2 \cdot \frac{-1}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   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 <= 0.25f) {
		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 <= 0.25e0) 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(0.25))
		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(0.25))
		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 < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in v around inf 68.4%

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

   6. Taylor expanded in u around 0 68.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \left(-2 \cdot \frac{u}{v} - 2 \cdot \frac{-1}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - 2 \cdot \frac{-1}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ 1.0 (- (* u (- 2.0 (* 2.0 (/ -1.0 v)))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f - (2.0f * (-1.0f / v)))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 - (2.0e0 * ((-1.0e0) / v)))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) - Float32(Float32(2.0) * Float32(Float32(-1.0) / v)))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) - (single(2.0) * (single(-1.0) / v)))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in v around inf 68.4%

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

   6. Taylor expanded in u around 0 67.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - 2 \cdot \frac{-1}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - 2 \cdot \frac{-1}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ -1.0 (* u (- 2.0 (* 2.0 (/ -1.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - (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 <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - (2.0e0 * ((-1.0e0) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - 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(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - (single(2.0) * (single(-1.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in v around inf 68.4%

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

   6. Taylor expanded in u around 0 67.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - 2 \cdot \frac{-1}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + u \cdot \left(2 - \frac{2}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25) 1.0 (+ 1.0 (* u (- 2.0 (/ 2.0 u))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (u * (2.0f - (2.0f / u)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (u * (2.0e0 - (2.0e0 / u)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(2.0) / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (u * (single(2.0) - (single(2.0) / u)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

      1. fma-undefine 90.2%

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

      2. fma-undefine 90.4%

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

      3. +-commutative 90.4%

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

      4. add-log-exp 90.4%

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

      5. *-commutative 90.4%

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

      6. exp-to-pow 89.8%

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

      7. +-commutative 89.8%

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

      8. fma-undefine 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in v around inf 55.0%

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

   8. Taylor expanded in u around inf 55.1%

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

      1. associate-*r/ 55.1%

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

      2. metadata-eval 55.1%

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

   10. Simplified 55.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + u \cdot \left(2 - \frac{2}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.2% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.25) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. fma-undefine 99.8%

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

      5. +-commutative 99.8%

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

      6. add-cube-cbrt 99.8%

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

      7. associate-*r* 99.7%

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

      8. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-define 90.1%

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

      3. +-commutative 90.1%

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

      4. fma-define 89.9%

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

   3. Simplified 89.9%

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

      1. fma-undefine 90.2%

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

      2. fma-undefine 90.4%

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

      3. +-commutative 90.4%

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

      4. add-log-exp 90.4%

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

      5. *-commutative 90.4%

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

      6. exp-to-pow 89.8%

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

      7. +-commutative 89.8%

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

      8. fma-undefine 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in v around inf 55.0%

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

   8. Taylor expanded in u around 0 55.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 99.2%

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

   1. +-commutative 99.2%

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

   2. +-commutative 99.2%

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

   3. fma-undefine 99.2%

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

   4. fma-undefine 99.2%

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

   5. +-commutative 99.2%

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

   6. add-cube-cbrt 99.2%

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

   7. associate-*r* 99.2%

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

   8. fma-define 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0 86.5%

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

Alternative 12: 5.8% 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 99.2%

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

   1. +-commutative 99.2%

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

   2. fma-define 99.2%

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

   3. +-commutative 99.2%

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

   4. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in u around 0 5.7%

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

Reproduce

herbie shell --seed 2024133 
(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))))))))