HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 15

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-define 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-define 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-undefine 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. Add Preprocessing

Alternative 3: 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 99.4%

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

3. Taylor expanded in u around 0 94.5%

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

Alternative 4: 91.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(2 - u \cdot 2\right) - \frac{\frac{u \cdot \left(0.5 \cdot \left(u \cdot 9.333333333333334 - \left(4 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 32\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - 1.3333333333333333\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (-
       (* u (- 2.0 (* u 2.0)))
       (/
        (+
         (/
          (*
           u
           (-
            (*
             0.5
             (-
              (* u 9.333333333333334)
              (+ (* 4.0 (- (* u 8.0) (* u 16.0))) (* u 32.0))))
            0.6666666666666666))
          v)
         (* u (- (* 0.5 (- (* u 16.0) (* u 8.0))) 1.3333333333333333)))
        v))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + (((u * (2.0f - (u * 2.0f))) - ((((u * ((0.5f * ((u * 9.333333333333334f) - ((4.0f * ((u * 8.0f) - (u * 16.0f))) + (u * 32.0f)))) - 0.6666666666666666f)) / v) + (u * ((0.5f * ((u * 16.0f) - (u * 8.0f))) - 1.3333333333333333f))) / 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 <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + (((u * (2.0e0 - (u * 2.0e0))) - ((((u * ((0.5e0 * ((u * 9.333333333333334e0) - ((4.0e0 * ((u * 8.0e0) - (u * 16.0e0))) + (u * 32.0e0)))) - 0.6666666666666666e0)) / v) + (u * ((0.5e0 * ((u * 16.0e0) - (u * 8.0e0))) - 1.3333333333333333e0))) / v)) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(u * Float32(Float32(2.0) - Float32(u * Float32(2.0)))) - Float32(Float32(Float32(Float32(u * Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(9.333333333333334)) - Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) + Float32(u * Float32(32.0))))) - Float32(0.6666666666666666))) / v) + Float32(u * Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - Float32(1.3333333333333333)))) / v)) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.1599999964237213))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + (((u * (single(2.0) - (u * single(2.0)))) - ((((u * ((single(0.5) * ((u * single(9.333333333333334)) - ((single(4.0) * ((u * single(8.0)) - (u * single(16.0)))) + (u * single(32.0))))) - single(0.6666666666666666))) / v) + (u * ((single(0.5) * ((u * single(16.0)) - (u * single(8.0)))) - single(1.3333333333333333)))) / v)) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in u around 0 66.4%

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

      1. fma-neg 66.4%

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

   7. Simplified 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\left(-0.5 \cdot \left(v \cdot u\right)\right) \cdot {\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{{\left(e^{\frac{-2}{v}}\right)}^{2}}\right), -1\right)} \]

   8. Taylor expanded in v around -inf 62.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(2 - u \cdot 2\right) - \frac{\frac{u \cdot \left(0.5 \cdot \left(u \cdot 9.333333333333334 - \left(4 \cdot \left(u \cdot 8 - u \cdot 16\right) + u \cdot 32\right)\right) - 0.6666666666666666\right)}{v} + u \cdot \left(0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - 1.3333333333333333\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(u \cdot 8 - u \cdot 16\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (/ (* u (+ 1.3333333333333333 (* 0.5 (- (* u 8.0) (* u 16.0))))) v)
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + ((((u * (1.3333333333333333f + (0.5f * ((u * 8.0f) - (u * 16.0f))))) / v) + (u * (2.0f - (u * 2.0f)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((((u * (1.3333333333333333e0 + (0.5e0 * ((u * 8.0e0) - (u * 16.0e0))))) / v) + (u * (2.0e0 - (u * 2.0e0)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(u * Float32(Float32(1.3333333333333333) + Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))))) / v) + Float32(u * Float32(Float32(2.0) - Float32(u * Float32(2.0))))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.1599999964237213))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + ((((u * (single(1.3333333333333333) + (single(0.5) * ((u * single(8.0)) - (u * single(16.0)))))) / v) + (u * (single(2.0) - (u * single(2.0))))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in u around 0 66.4%

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

      1. fma-neg 66.4%

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

   7. Simplified 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\left(-0.5 \cdot \left(v \cdot u\right)\right) \cdot {\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{{\left(e^{\frac{-2}{v}}\right)}^{2}}\right), -1\right)} \]

   8. Taylor expanded in v around -inf 58.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(u \cdot 8 - u \cdot 16\right)\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+
    1.0
    (-
     (/
      (-
       (/ (+ (* 0.6666666666666666 (/ u v)) (* u 1.3333333333333333)) v)
       (* u -2.0))
      v)
     (+ 2.0 (* u -2.0))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((((((0.6666666666666666f * (u / v)) + (u * 1.3333333333333333f)) / v) - (u * -2.0f)) / v) - (2.0f + (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 <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((((((0.6666666666666666e0 * (u / v)) + (u * 1.3333333333333333e0)) / v) - (u * (-2.0e0))) / v) - (2.0e0 + (u * (-2.0e0))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u / v)) + Float32(u * Float32(1.3333333333333333))) / v) - Float32(u * Float32(-2.0))) / v) - Float32(Float32(2.0) + Float32(u * Float32(-2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.1599999964237213))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((((((single(0.6666666666666666) * (u / v)) + (u * single(1.3333333333333333))) / v) - (u * single(-2.0))) / v) - (single(2.0) + (u * single(-2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. flip-- 92.9%

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

      2. associate-*l/ 93.3%

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

      3. metadata-eval 93.3%

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

      4. pow2 93.3%

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in u around 0 55.6%

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

   6. Taylor expanded in v around -inf 56.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.6666666666666666 \cdot \frac{u}{v} + u \cdot 1.3333333333333333}{v} - u \cdot -2}{v} - \left(2 + u \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   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 <= 0.1599999964237213f) {
		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 <= 0.1599999964237213e0) 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(0.1599999964237213))
		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(0.1599999964237213))
		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 < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in v around inf 54.8%

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

   6. Taylor expanded in u around 0 54.8%

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

4. Final simplification 87.5%

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

Alternative 8: 90.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   1.0
   (+ -1.0 (* u (+ 2.0 (/ (+ 2.0 (* 1.3333333333333333 (/ 1.0 v))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (1.3333333333333333f * (1.0f / 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 <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 * (1.0e0 / v))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.1599999964237213))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) * Float32(Float32(1.0) / v))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.1599999964237213))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (single(1.3333333333333333) * (single(1.0) / v))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in v around -inf 64.4%

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

   6. Taylor expanded in u around 0 55.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + 1.3333333333333333 \cdot \frac{1}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.8% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.1599999964237213e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((u * (2.0e0 + (u * (-2.0e0)))) / v))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in u around 0 66.4%

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

      1. fma-neg 66.4%

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

   7. Simplified 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\left(-0.5 \cdot \left(v \cdot u\right)\right) \cdot {\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{{\left(e^{\frac{-2}{v}}\right)}^{2}}\right), -1\right)} \]

   8. Taylor expanded in v around inf 54.8%

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

4. Final simplification 87.5%

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

Alternative 10: 90.7% accurate, 11.8× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.1599999964237213)
   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 <= 0.1599999964237213f) {
		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 <= 0.1599999964237213e0) 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(0.1599999964237213))
		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(0.1599999964237213))
		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 < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in v around inf 54.8%

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

   6. Taylor expanded in u around 0 52.8%

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

Alternative 11: 90.7% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. flip-- 92.9%

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

      2. associate-*l/ 93.3%

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

      3. metadata-eval 93.3%

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

      4. pow2 93.3%

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in u around 0 55.6%

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

   6. Taylor expanded in v around inf 52.8%

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

4. Final simplification 87.3%

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

Alternative 12: 90.7% accurate, 13.3× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + (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 <= 0.1599999964237213e0) 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(0.1599999964237213))
		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(0.1599999964237213))
		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 < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in v around inf 54.8%

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

   6. Taylor expanded in u around 0 52.8%

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

4. Final simplification 87.3%

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

Alternative 13: 90.1% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.1599999964237213f) {
		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 <= 0.1599999964237213e0) 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(0.1599999964237213))
		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(0.1599999964237213))
		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 < 0.159999996

   1. Initial program 100.0%

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

      1. log1p-expm1-u 100.0%

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

      2. expm1-undefine 99.9%

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

      3. *-commutative 99.9%

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

      4. exp-to-pow 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 90.9%

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

   if 0.159999996 < v

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. fma-define 94.1%

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

      3. +-commutative 94.1%

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

      4. fma-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in u around 0 66.4%

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

      1. fma-neg 66.4%

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

   7. Simplified 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(v, \mathsf{expm1}\left(\frac{2}{v}\right), \frac{\left(-0.5 \cdot \left(v \cdot u\right)\right) \cdot {\left(1 - e^{\frac{-2}{v}}\right)}^{2}}{{\left(e^{\frac{-2}{v}}\right)}^{2}}\right), -1\right)} \]

   8. Taylor expanded in v around inf 44.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.1599999964237213:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 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 99.4%

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

   1. log1p-expm1-u 99.4%

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

   2. expm1-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. exp-to-pow 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-undefine 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in v around 0 83.0%

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

Alternative 15: 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. 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-define 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-define 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. Taylor expanded in u around 0 6.3%

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

Reproduce

herbie shell --seed 2024137 
(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))))))))