HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 16.9s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-def 99.4%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 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 99.4%

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

3. Final simplification 99.4%

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

Alternative 3: 91.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \left(\frac{\frac{u \cdot 1.3333333333333333}{v}}{v} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + \left(1 - u\right) \cdot 4}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    1.0
    (+
     (* (- 1.0 u) -2.0)
     (+
      (/ (/ (* u 1.3333333333333333) v) v)
      (* 0.5 (/ (+ (* -4.0 (pow (- 1.0 u) 2.0)) (* (- 1.0 u) 4.0)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + ((((u * 1.3333333333333333f) / v) / v) + (0.5f * (((-4.0f * powf((1.0f - 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + ((((u * 1.3333333333333333e0) / v) / v) + (0.5e0 * ((((-4.0e0) * ((1.0e0 - 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(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(Float32(Float32(u * Float32(1.3333333333333333)) / v) / v) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * (Float32(Float32(1.0) - 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(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + ((((u * single(1.3333333333333333)) / v) / v) + (single(0.5) * (((single(-4.0) * ((single(1.0) - 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 < 0.119999997

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.1%

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

   if 0.119999997 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in v around inf 73.0%

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

   4. Taylor expanded in u around 0 70.2%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{8 \cdot u}}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
   5. Step-by-step derivation

      1. *-commutative 70.2%

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

   6. Simplified 70.2%

      \[\leadsto 1 + \left(-2 \cdot \left(1 - u\right) + \left(0.16666666666666666 \cdot \frac{\color{blue}{u \cdot 8}}{{v}^{2}} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)}{v}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 70.2%

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

      2. unpow2 70.2%

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

      3. associate-/r* 70.2%

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

      4. *-commutative 70.2%

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

      5. associate-*r* 70.2%

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

      6. metadata-eval 70.2%

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

   8. Applied egg-rr 70.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + \left(\frac{\frac{u \cdot 1.3333333333333333}{v}}{v} + 0.5 \cdot \frac{-4 \cdot {\left(1 - u\right)}^{2} + \left(1 - u\right) \cdot 4}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1.3333333333333333 \cdot \frac{u}{{v}^{2}} + \left(u \cdot 2 + 2 \cdot \frac{u}{v}\right)\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    1.0
    (-
     (+ (* 1.3333333333333333 (/ u (pow v 2.0))) (+ (* u 2.0) (* 2.0 (/ u v))))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.3333333333333333f * (u / powf(v, 2.0f))) + ((u * 2.0f) + (2.0f * (u / 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 <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.3333333333333333e0 * (u / (v ** 2.0e0))) + ((u * 2.0e0) + (2.0e0 * (u / v)))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.3333333333333333) * Float32(u / (v ^ Float32(2.0)))) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(2.0) * Float32(u / v)))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.3333333333333333) * (u / (v ^ single(2.0)))) + ((u * single(2.0)) + (single(2.0) * (u / v)))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.1%

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

   if 0.119999997 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 63.6%

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

   4. Taylor expanded in v around inf 66.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1.3333333333333333 \cdot \frac{u}{{v}^{2}} + \left(u \cdot 2 + 2 \cdot \frac{u}{v}\right)\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u \cdot 2 + 2 \cdot \frac{u}{v}\right) + 1.3333333333333333 \cdot \left(\frac{u}{v} \cdot \mathsf{reciprocal}\left(v\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+
    (+
     (+ (* u 2.0) (* 2.0 (/ u v)))
     (* 1.3333333333333333 (* (/ u v) (reciprocal v))))
    -1.0)))

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.1%

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

   if 0.119999997 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 63.6%

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

      1. cancel-sign-sub-inv 63.6%

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

      2. rec-exp 63.6%

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

      3. expm1-def 63.6%

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

      4. metadata-eval 63.6%

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

      5. div-inv 63.6%

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

   5. Applied egg-rr 63.6%

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

   6. Taylor expanded in v around inf 66.5%

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

      1. *-un-lft-identity 66.5%

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

      2. unpow2 66.5%

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

      3. times-frac 66.5%

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

      4. reciprocal-define 66.5%

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

   8. Applied egg-rr 66.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u \cdot 2 + 2 \cdot \frac{u}{v}\right) + 1.3333333333333333 \cdot \left(\frac{u}{v} \cdot \mathsf{reciprocal}\left(v\right)\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{1.3333333333333333}{{v}^{2}} + \frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.11999999731779099)
   1.0
   (+ (* u (+ 2.0 (+ (/ 1.3333333333333333 (pow v 2.0)) (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.11999999731779099f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (2.0f + ((1.3333333333333333f / powf(v, 2.0f)) + (2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.11999999731779099e0) then
        tmp = 1.0e0
    else
        tmp = (u * (2.0e0 + ((1.3333333333333333e0 / (v ** 2.0e0)) + (2.0e0 / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.11999999731779099))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) / (v ^ Float32(2.0))) + Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.11999999731779099))
		tmp = single(1.0);
	else
		tmp = (u * (single(2.0) + ((single(1.3333333333333333) / (v ^ single(2.0))) + (single(2.0) / v)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.119999997

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 93.1%

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

   if 0.119999997 < v

   1. Initial program 92.3%

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

   3. Taylor expanded in u around 0 63.6%

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

      1. cancel-sign-sub-inv 63.6%

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

      2. rec-exp 63.6%

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

      3. expm1-def 63.6%

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

      4. metadata-eval 63.6%

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

      5. div-inv 63.6%

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

   5. Applied egg-rr 63.6%

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

   6. Taylor expanded in v around inf 66.5%

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

   7. Taylor expanded in u around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. metadata-eval 66.5%

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

      3. associate-*r/ 66.5%

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

      4. metadata-eval 66.5%

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

   9. Simplified 66.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.11999999731779099:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 + \left(\frac{1.3333333333333333}{{v}^{2}} + \frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (+ (* u (* v (+ (exp (/ 2.0 v)) -1.0))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = (u * (v * (expf((2.0f / v)) + -1.0f))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = (u * (v * (exp((2.0e0 / v)) + (-1.0e0)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(u * Float32(v * Float32(exp(Float32(Float32(2.0) / v)) + Float32(-1.0)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = (u * (v * (exp((single(2.0) / v)) + single(-1.0)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.400000006 < v

   1. Initial program 91.3%

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

   3. Taylor expanded in u around 0 77.8%

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

      1. cancel-sign-sub-inv 77.8%

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

      2. rec-exp 77.8%

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

      3. expm1-def 77.8%

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

      4. metadata-eval 77.8%

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

      5. div-inv 77.8%

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

   5. Applied egg-rr 77.8%

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

   6. Taylor expanded in v around 0 79.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(e^{\frac{2}{v}} + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645) 1.0 (+ (* v (* u (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = (v * (u * expm1f((2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(v * Float32(u * expm1(Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.8%

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

   if 0.400000006 < v

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. fma-def 91.4%

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

      3. +-commutative 91.4%

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

      4. fma-def 90.7%

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

   3. Simplified 90.7%

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

      1. fma-udef 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. *-un-lft-identity 91.4%

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

      2. exp-prod 90.8%

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

   8. Applied egg-rr 90.8%

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

      1. exp-1-e 90.8%

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

   10. Simplified 90.8%

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

   11. Taylor expanded in u around 0 77.1%

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

       1. associate-+r+ 76.8%

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

       2. log-E 79.1%

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

       3. metadata-eval 79.1%

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

       4. metadata-eval 79.1%

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

       5. associate-*r* 79.1%

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

       6. rec-exp 79.1%

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

       7. log-E 79.1%

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

       8. reciprocal-define 78.7%

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

       9. distribute-lft-neg-in 78.7%

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

       10. metadata-eval 78.7%

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

       11. reciprocal-define 79.1%

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

       12. associate-*r/ 79.1%

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

       13. metadata-eval 79.1%

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

       14. expm1-def 79.1%

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

   13. Simplified 71.9%

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

   14. Taylor expanded in u around 0 79.1%

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

   16. Simplified 79.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.5% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.150000006 < v

   1. Initial program 91.9%

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

   3. Taylor expanded in u around 0 66.2%

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

   4. Taylor expanded in v around inf 65.6%

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

      1. sub-neg 65.6%

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

      2. distribute-lft-out 65.6%

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

      3. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

4. Final simplification 90.9%

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

Alternative 10: 90.0% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = ((2.0e0 / v) * (v * u)) + (-1.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 92.2%

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

   if 0.219999999 < v

   1. Initial program 91.5%

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

      1. +-commutative 91.5%

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

      2. fma-def 91.6%

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

      3. +-commutative 91.6%

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

      4. fma-def 90.9%

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

   3. Simplified 90.9%

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

      1. fma-udef 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. *-un-lft-identity 91.6%

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

      2. exp-prod 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. exp-1-e 91.1%

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

   10. Simplified 91.1%

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

   11. Taylor expanded in u around 0 72.5%

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

       1. associate-+r+ 72.2%

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

       2. log-E 74.3%

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

       3. metadata-eval 74.3%

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

       4. metadata-eval 74.3%

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

       5. associate-*r* 74.3%

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

       6. rec-exp 74.3%

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

       7. log-E 74.3%

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

       8. reciprocal-define 74.0%

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

       9. distribute-lft-neg-in 74.0%

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

       10. metadata-eval 74.0%

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

       11. reciprocal-define 74.3%

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

       12. associate-*r/ 74.3%

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

       13. metadata-eval 74.3%

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

       14. expm1-def 74.3%

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

   13. Simplified 67.6%

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

   14. Taylor expanded in v around inf 59.2%

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

4. Final simplification 90.2%

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

Alternative 11: 90.5% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.150000006 < v

   1. Initial program 91.9%

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

   3. Taylor expanded in u around 0 66.2%

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

      1. cancel-sign-sub-inv 66.2%

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

      2. rec-exp 66.2%

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

      3. expm1-def 66.2%

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

      4. metadata-eval 66.2%

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

      5. div-inv 66.2%

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

   5. Applied egg-rr 66.2%

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

   6. Taylor expanded in v around inf 65.5%

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

      1. sub-neg 65.5%

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

      2. distribute-lft-out 65.5%

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

      3. metadata-eval 65.5%

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

   8. Simplified 65.5%

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

4. Final simplification 90.9%

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

Alternative 12: 90.0% accurate, 21.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.2199999988079071f) {
		tmp = 1.0f;
	} else {
		tmp = (u * 2.0f) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.2199999988079071e0) then
        tmp = 1.0e0
    else
        tmp = (u * 2.0e0) + (-1.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.219999999

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 92.2%

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

   if 0.219999999 < v

   1. Initial program 91.5%

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

   3. Taylor expanded in v around inf 59.2%

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

   4. Taylor expanded in u around 0 59.2%

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

4. Final simplification 90.2%

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

Alternative 13: 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 99.4%

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

3. Taylor expanded in u around 0 5.9%

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

4. Final simplification 5.9%

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

Alternative 14: 87.0% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-def 99.4%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in v around 0 86.8%

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

8. Final simplification 86.8%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))))