HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.0s

Alternatives: 24

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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) \]
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 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.5%

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

3. Taylor expanded in u around 0 95.7%

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

Alternative 4: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{-0.5 \cdot \frac{u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (+
      (/
       (-
        (/
         (+
          (*
           -0.5
           (/
            (+
             (* u 9.333333333333334)
             (- (* 4.0 (- (* u 16.0) (* u 8.0))) (* u 32.0)))
            v))
          (* (- (* u 8.0) (* u 16.0)) 0.5))
         v)
        (* u 2.0))
       v)
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((((((-0.5f * (((u * 9.333333333333334f) + ((4.0f * ((u * 16.0f) - (u * 8.0f))) - (u * 32.0f))) / v)) + (((u * 8.0f) - (u * 16.0f)) * 0.5f)) / v) - (u * 2.0f)) / v) + (v * (-1.0f + (1.0f / expf((-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 <= 0.30000001192092896e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * (((((((-0.5e0) * (((u * 9.333333333333334e0) + ((4.0e0 * ((u * 16.0e0) - (u * 8.0e0))) - (u * 32.0e0))) / v)) + (((u * 8.0e0) - (u * 16.0e0)) * 0.5e0)) / v) - (u * 2.0e0)) / v) + (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v)))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - Float32(u * Float32(32.0)))) / v)) + Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5))) / v) - Float32(u * Float32(2.0))) / v) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((((((single(-0.5) * (((u * single(9.333333333333334)) + ((single(4.0) * ((u * single(16.0)) - (u * single(8.0)))) - (u * single(32.0)))) / v)) + (((u * single(8.0)) - (u * single(16.0))) * single(0.5))) / v) - (u * single(2.0))) / v) + (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.6%

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

   4. Taylor expanded in u around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. log-rec 99.4%

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

      4. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   if 0.300000012 < v

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. fma-define 94.0%

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

      3. +-commutative 94.0%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 87.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{-0.5 \cdot \frac{u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) + -0.5 \cdot \frac{u \cdot 4 + 8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (+
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
      (* -0.5 (/ (+ (* u 4.0) (* 8.0 (/ u v))) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) + (-0.5f * (((u * 4.0f) + (8.0f * (u / v))) / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) + ((-0.5e0) * (((u * 4.0e0) + (8.0e0 * (u / v))) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) + Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(4.0)) + Float32(Float32(8.0) * Float32(u / v))) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) + (single(-0.5) * (((u * single(4.0)) + (single(8.0) * (u / v))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.6%

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

   4. Taylor expanded in u around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. log-rec 99.4%

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

      4. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   if 0.300000012 < v

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. fma-define 94.0%

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

      3. +-commutative 94.0%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around inf 85.5%

      \[\leadsto u \cdot \left(-0.5 \cdot \color{blue}{\frac{\left(-8 \cdot \frac{u}{v} + 4 \cdot u\right) - -16 \cdot \frac{u}{v}}{v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 85.5%

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

      2. +-commutative 85.5%

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

      3. associate-+l+ 85.5%

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

      4. *-commutative 85.5%

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

      5. sub-neg 85.5%

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

      6. distribute-rgt-out-- 85.5%

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

      7. metadata-eval 85.5%

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

   8. Simplified 85.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) + -0.5 \cdot \frac{u \cdot 4 + 8 \cdot \frac{u}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) + -2 \cdot \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (* u (+ (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))) (* -2.0 (/ u v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) + (-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 <= 0.30000001192092896e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) + ((-2.0e0) * (u / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) + 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(0.30000001192092896))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) + (single(-2.0) * (u / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.6%

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

   4. Taylor expanded in u around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. log-rec 99.4%

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

      4. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   if 0.300000012 < v

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. fma-define 94.0%

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

      3. +-commutative 94.0%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around inf 83.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) + -2 \cdot \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log u)))
   (+ -1.0 (* u (* v (expm1 (/ 2.0 v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * (v * expm1f((2.0f / v))));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(v * expm1(Float32(Float32(2.0) / v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.6%

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

   4. Taylor expanded in u around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. log-rec 99.4%

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

      4. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   if 0.300000012 < v

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. fma-define 94.0%

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

      3. +-commutative 94.0%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in u around 0 79.5%

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

      1. rec-exp 79.5%

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

      2. expm1-undefine 79.5%

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

      3. distribute-neg-frac 79.5%

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

      4. metadata-eval 79.5%

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

   8. Simplified 79.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (+ 1.0 (* v (log u)))
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/
          (+
           1.3333333333333333
           (+
            (* (- (* u 8.0) (* u 16.0)) 0.5)
            (/
             (-
              0.6666666666666666
              (*
               (+
                (* u 9.333333333333334)
                (- (* 4.0 (- (* u 16.0) (* u 8.0))) (* u 32.0)))
               0.5))
             v)))
          v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f + ((((u * 8.0f) - (u * 16.0f)) * 0.5f) + ((0.6666666666666666f - (((u * 9.333333333333334f) + ((4.0f * ((u * 16.0f) - (u * 8.0f))) - (u * 32.0f))) * 0.5f)) / v))) / v) - (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 <= 0.30000001192092896e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + ((((u * 8.0e0) - (u * 16.0e0)) * 0.5e0) + ((0.6666666666666666e0 - (((u * 9.333333333333334e0) + ((4.0e0 * ((u * 16.0e0) - (u * 8.0e0))) - (u * 32.0e0))) * 0.5e0)) / v))) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5)) + Float32(Float32(Float32(0.6666666666666666) - Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - Float32(u * Float32(32.0)))) * Float32(0.5))) / v))) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((((u * single(8.0)) - (u * single(16.0))) * single(0.5)) + ((single(0.6666666666666666) - (((u * single(9.333333333333334)) + ((single(4.0) * ((u * single(16.0)) - (u * single(8.0)))) - (u * single(32.0)))) * single(0.5))) / v))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.6%

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

   4. Taylor expanded in u around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. log-rec 99.4%

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

      4. remove-double-neg 99.4%

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

   6. Simplified 99.4%

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

   if 0.300000012 < v

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. fma-define 94.0%

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

      3. +-commutative 94.0%

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

      4. fma-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in u around 0 89.4%

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

   6. Taylor expanded in v around -inf 78.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\log e\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (log E)
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/
          (+
           1.3333333333333333
           (+
            (* (- (* u 8.0) (* u 16.0)) 0.5)
            (/
             (-
              0.6666666666666666
              (*
               (+
                (* u 9.333333333333334)
                (- (* 4.0 (- (* u 16.0) (* u 8.0))) (* u 32.0)))
               0.5))
             v)))
          v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = logf(((float) M_E));
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f + ((((u * 8.0f) - (u * 16.0f)) * 0.5f) + ((0.6666666666666666f - (((u * 9.333333333333334f) + ((4.0f * ((u * 16.0f) - (u * 8.0f))) - (u * 32.0f))) * 0.5f)) / v))) / v) - (u * 2.0f))) / v)));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = log(Float32(exp(1)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5)) + Float32(Float32(Float32(0.6666666666666666) - Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - Float32(u * Float32(32.0)))) * Float32(0.5))) / v))) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = log(single(2.71828182845904523536));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((((u * single(8.0)) - (u * single(16.0))) * single(0.5)) + ((single(0.6666666666666666) - (((u * single(9.333333333333334)) + ((single(4.0) * ((u * single(16.0)) - (u * single(8.0)))) - (u * single(32.0)))) * single(0.5))) / v))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. log1p-expm1-u 99.9%

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

      2. log1p-undefine 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in v around 0 91.4%

      \[\leadsto \log \color{blue}{\left(e^{1}\right)} \]
   8. Step-by-step derivation

      1. exp-1-e 91.4%

         \[\leadsto \log \color{blue}{e} \]

   9. Simplified 91.4%

      \[\leadsto \log \color{blue}{e} \]

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in v around -inf 74.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;\log e\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/
          (+
           1.3333333333333333
           (+
            (* (- (* u 8.0) (* u 16.0)) 0.5)
            (/
             (-
              0.6666666666666666
              (*
               (+
                (* u 9.333333333333334)
                (- (* 4.0 (- (* u 16.0) (* u 8.0))) (* u 32.0)))
               0.5))
             v)))
          v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f + ((((u * 8.0f) - (u * 16.0f)) * 0.5f) + ((0.6666666666666666f - (((u * 9.333333333333334f) + ((4.0f * ((u * 16.0f) - (u * 8.0f))) - (u * 32.0f))) * 0.5f)) / v))) / v) - (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + ((((u * 8.0e0) - (u * 16.0e0)) * 0.5e0) + ((0.6666666666666666e0 - (((u * 9.333333333333334e0) + ((4.0e0 * ((u * 16.0e0) - (u * 8.0e0))) - (u * 32.0e0))) * 0.5e0)) / v))) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5)) + Float32(Float32(Float32(0.6666666666666666) - Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - Float32(u * Float32(32.0)))) * Float32(0.5))) / v))) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((((u * single(8.0)) - (u * single(16.0))) * single(0.5)) + ((single(0.6666666666666666) - (((u * single(9.333333333333334)) + ((single(4.0) * ((u * single(16.0)) - (u * single(8.0)))) - (u * single(32.0)))) * single(0.5))) / v))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in v around -inf 74.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{u \cdot \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + 1.3333333333333333\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (/ (* u (+ (* (- (* u 8.0) (* u 16.0)) 0.5) 1.3333333333333333)) v)
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + ((u * 2.0f) + ((((u * ((((u * 8.0f) - (u * 16.0f)) * 0.5f) + 1.3333333333333333f)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + ((((u * ((((u * 8.0e0) - (u * 16.0e0)) * 0.5e0) + 1.3333333333333333e0)) / 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(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(Float32(u * Float32(Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5)) + Float32(1.3333333333333333))) / 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(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + ((u * single(2.0)) + ((((u * ((((u * single(8.0)) - (u * single(16.0))) * single(0.5)) + single(1.3333333333333333))) / 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 < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in v around -inf 70.1%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{\frac{u \cdot \left(\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + 1.3333333333333333\right)}{v} + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + 1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/ (+ (* (- (* u 8.0) (* u 16.0)) 0.5) 1.3333333333333333) v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((((((u * 8.0f) - (u * 16.0f)) * 0.5f) + 1.3333333333333333f) / v) - (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((((((u * 8.0e0) - (u * 16.0e0)) * 0.5e0) + 1.3333333333333333e0) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5)) + Float32(1.3333333333333333)) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((((((u * single(8.0)) - (u * single(16.0))) * single(0.5)) + single(1.3333333333333333)) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in v around -inf 70.1%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 + 1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) v))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f * (1.0f / v))) / v)) / 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in u around 0 75.0%

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

   4. Taylor expanded in v around -inf 69.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+ 2.0 (/ (+ 1.3333333333333333 (* 0.6666666666666666 (/ 1.0 v))) v))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f + (0.6666666666666666f * (1.0f / v))) / v)) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 + (0.6666666666666666e0 * (1.0e0 / v))) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(0.6666666666666666) * Float32(Float32(1.0) / v))) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) + (single(0.6666666666666666) * (single(1.0) / v))) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in u around 0 75.4%

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

      1. rec-exp 75.4%

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

      2. expm1-undefine 75.4%

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

      3. distribute-neg-frac 75.4%

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

      4. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in v around -inf 69.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.4% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + u \cdot \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+ -1.0 (+ (* u 2.0) (* u (/ (+ 2.0 (/ 1.3333333333333333 v)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + ((u * 2.0f) + (u * ((2.0f + (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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + (u * ((2.0e0 + (1.3333333333333333e0 / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + ((u * single(2.0)) + (u * ((single(2.0) + (single(1.3333333333333333) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

      1. log1p-expm1-u 94.0%

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

      2. log1p-undefine 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in u around 0 75.5%

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

   8. Taylor expanded in v around -inf 67.1%

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

   9. Taylor expanded in u around 0 67.1%

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

       1. associate-/l* 67.1%

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

       2. associate-*r/ 67.1%

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

       3. metadata-eval 67.1%

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

   11. Simplified 67.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + u \cdot \frac{2 + \frac{1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + u \cdot -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+ -1.0 (* u (+ 2.0 (/ (+ 2.0 (* u -2.0)) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + (u * (2.0f + ((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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (u * (-2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(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(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (u * single(-2.0))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in v around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. associate-/l* 62.8%

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

      3. distribute-lft-out 62.8%

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

   8. Simplified 62.8%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + u \cdot -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+ -1.0 (* u (+ 2.0 (/ 2.0 v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = -1.0f + (u * (2.0f + (2.0f / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + (2.0e0 / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(2.0) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + (u * (single(2.0) + (single(2.0) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 84.1%

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

   6. Taylor expanded in u around 0 75.4%

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

      1. rec-exp 75.4%

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

      2. expm1-undefine 75.4%

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

      3. distribute-neg-frac 75.4%

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

      4. metadata-eval 75.4%

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

   8. Simplified 75.4%

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

   9. Taylor expanded in v around inf 62.0%

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

       1. associate-*r/ 62.0%

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

       2. metadata-eval 62.0%

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

   11. Simplified 62.0%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (- 1.0 (* u -2.0))
   (+ -1.0 (* 2.0 (+ u (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    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(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	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(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	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 < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

      1. log1p-expm1-u 94.0%

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

      2. log1p-undefine 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in u around 0 75.5%

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

   8. Taylor expanded in v around inf 62.0%

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

      1. sub-neg 62.0%

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

      2. distribute-lft-out 62.0%

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

      3. metadata-eval 62.0%

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

   10. Simplified 62.0%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.7% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) (- 1.0 (* u -2.0)) (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = 1.0f + ((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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in v around inf 54.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 49.7% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 - \frac{1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) (- 1.0 (* u -2.0)) (* u (- 2.0 (/ 1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} else {
		tmp = u * (2.0f - (1.0f / u));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    else
        tmp = u * (2.0e0 - (1.0e0 / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	else
		tmp = Float32(u * Float32(Float32(2.0) - Float32(Float32(1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = u * (single(2.0) - (single(1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around -inf 31.2%

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

      1. associate-*r/ 31.2%

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

      2. cancel-sign-sub-inv 31.2%

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

      3. metadata-eval 31.2%

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

      4. associate-*r/ 31.2%

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

      5. metadata-eval 31.2%

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

      6. distribute-lft-in 31.2%

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

      7. metadata-eval 31.2%

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

      8. neg-mul-1 31.2%

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

      9. distribute-neg-frac 31.2%

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

      10. metadata-eval 31.2%

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

   5. Simplified 31.2%

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

   6. Taylor expanded in v around inf 54.2%

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

      1. *-commutative 54.2%

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

   8. Simplified 54.2%

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

   9. Taylor expanded in u around inf 54.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(2 - \frac{1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 49.7% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) (- 1.0 (* u -2.0)) (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f - (u * -2.0f);
	} 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 <= 0.20000000298023224e0) then
        tmp = 1.0e0 - (u * (-2.0e0))
    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(0.20000000298023224))
		tmp = Float32(Float32(1.0) - Float32(u * Float32(-2.0)));
	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(0.20000000298023224))
		tmp = single(1.0) - (u * single(-2.0));
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 49.8%

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

       1. neg-mul-1 49.8%

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

   11. Simplified 49.8%

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

   if 0.200000003 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around -inf 31.2%

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

      1. associate-*r/ 31.2%

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

      2. cancel-sign-sub-inv 31.2%

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

      3. metadata-eval 31.2%

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

      4. associate-*r/ 31.2%

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

      5. metadata-eval 31.2%

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

      6. distribute-lft-in 31.2%

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

      7. metadata-eval 31.2%

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

      8. neg-mul-1 31.2%

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

      9. distribute-neg-frac 31.2%

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

      10. metadata-eval 31.2%

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

   5. Simplified 31.2%

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

   6. Taylor expanded in v around inf 54.2%

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

      1. *-commutative 54.2%

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

   8. Simplified 54.2%

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

   9. Taylor expanded in u around 0 54.2%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 - u \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.7% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) (* u 2.0) (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = u * 2.0f;
	} 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 <= 0.20000000298023224e0) then
        tmp = u * 2.0e0
    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(0.20000000298023224))
		tmp = Float32(u * Float32(2.0));
	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(0.20000000298023224))
		tmp = u * single(2.0);
	else
		tmp = single(-1.0) + (u * single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 21.1%

      \[\leadsto \color{blue}{2 \cdot u} \]
   10. Step-by-step derivation

       1. *-commutative 21.1%

          \[\leadsto \color{blue}{u \cdot 2} \]

   11. Simplified 21.1%

       \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.200000003 < v

   1. Initial program 94.0%

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

   3. Taylor expanded in v around -inf 31.2%

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

      1. associate-*r/ 31.2%

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

      2. cancel-sign-sub-inv 31.2%

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

      3. metadata-eval 31.2%

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

      4. associate-*r/ 31.2%

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

      5. metadata-eval 31.2%

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

      6. distribute-lft-in 31.2%

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

      7. metadata-eval 31.2%

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

      8. neg-mul-1 31.2%

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

      9. distribute-neg-frac 31.2%

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

      10. metadata-eval 31.2%

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

   5. Simplified 31.2%

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

   6. Taylor expanded in v around inf 54.2%

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

      1. *-commutative 54.2%

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

   8. Simplified 54.2%

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

   9. Taylor expanded in u around 0 54.2%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 22.1% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;u \cdot 2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) (* u 2.0) -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = u * 2.0f;
	} else {
		tmp = -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 <= 0.20000000298023224e0) then
        tmp = u * 2.0e0
    else
        tmp = -1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(u * Float32(2.0));
	else
		tmp = Float32(-1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = u * single(2.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. Taylor expanded in v around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. associate-*r/ 47.2%

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

      5. metadata-eval 47.2%

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

      6. distribute-lft-in 47.2%

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

      7. metadata-eval 47.2%

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

      8. neg-mul-1 47.2%

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

      9. distribute-neg-frac 47.2%

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

      10. metadata-eval 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in v around inf 4.8%

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

      1. *-commutative 4.8%

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

   8. Simplified 4.8%

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

   9. Taylor expanded in u around inf 21.1%

      \[\leadsto \color{blue}{2 \cdot u} \]
   10. Step-by-step derivation

       1. *-commutative 21.1%

          \[\leadsto \color{blue}{u \cdot 2} \]

   11. Simplified 21.1%

       \[\leadsto \color{blue}{u \cdot 2} \]

   if 0.200000003 < v

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. fma-define 94.3%

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

      3. +-commutative 94.3%

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

      4. fma-define 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in u around 0 45.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 5.6% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u around 0 6.6%

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

Reproduce

herbie shell --seed 2024125 
(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))))))))