HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.9s

Alternatives: 18

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 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) \]
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 99.6%

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

Alternative 3: 95.7% 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 99.6%

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

3. Taylor expanded in u around 0 96.5%

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

Alternative 4: 94.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in u around 0 96.5%

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

8. Taylor expanded in u around inf 95.1%

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

9. Final simplification 95.1%

   \[\leadsto 1 - v \cdot \log \left(\frac{1}{u}\right) \]
10. Add Preprocessing

Alternative 5: 94.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log u \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (+ 1.0 (* v (log u))))

C

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

Fortran

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

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(u)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in u around 0 96.5%

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

8. Taylor expanded in u around inf 95.1%

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

   1. mul-1-neg 95.1%

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

   2. distribute-rgt-neg-in 95.1%

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

   3. log-rec 95.1%

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

   4. remove-double-neg 95.1%

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

10. Simplified 95.1%

    \[\leadsto \color{blue}{1 + v \cdot \log u} \]
11. Add Preprocessing

Alternative 6: 91.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - \left(\frac{0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - 1.3333333333333333}{v} + u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (-
        2.0
        (+
         (/ (- (* 0.5 (- (* u 16.0) (* u 8.0))) 1.3333333333333333) v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f - ((((0.5f * ((u * 16.0f) - (u * 8.0f))) - 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 - ((((0.5e0 * ((u * 16.0e0) - (u * 8.0e0))) - 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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) - Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) - ((((single(0.5) * ((u * single(16.0)) - (u * single(8.0)))) - single(1.3333333333333333)) / v) + (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 - \left(\frac{0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - 1.3333333333333333}{v} + u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{\frac{u \cdot \left(v \cdot \left(u \cdot 2 - 2\right)\right) + u \cdot \left(u \cdot 4 - 1.3333333333333333\right)}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (-
     (* u 2.0)
     (/
      (/
       (+ (* u (* v (- (* u 2.0) 2.0))) (* u (- (* u 4.0) 1.3333333333333333)))
       v)
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - ((((u * (v * ((u * 2.0f) - 2.0f))) + (u * ((u * 4.0f) - 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - ((((u * (v * ((u * 2.0e0) - 2.0e0))) + (u * ((u * 4.0e0) - 1.3333333333333333e0))) / v) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(u * Float32(v * Float32(Float32(u * Float32(2.0)) - Float32(2.0)))) + Float32(u * Float32(Float32(u * Float32(4.0)) - Float32(1.3333333333333333)))) / v) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - ((((u * (v * ((u * single(2.0)) - single(2.0)))) + (u * ((u * single(4.0)) - single(1.3333333333333333)))) / v) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 59.3%

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

      1. associate-*r/ 59.3%

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

      2. metadata-eval 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in v around 0 59.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{\frac{u \cdot \left(v \cdot \left(u \cdot 2 - 2\right)\right) + u \cdot \left(u \cdot 4 - 1.3333333333333333\right)}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(\frac{1.3333333333333333}{v} - 4 \cdot \frac{u}{v}\right) + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/
      (+
       (* u (- (/ 1.3333333333333333 v) (* 4.0 (/ u v))))
       (* u (- 2.0 (* u 2.0))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + (((u * ((1.3333333333333333f / v) - (4.0f * (u / 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + (((u * ((1.3333333333333333e0 / v) - (4.0e0 * (u / 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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(u * Float32(Float32(Float32(1.3333333333333333) / v) - Float32(Float32(4.0) * Float32(u / 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + (((u * ((single(1.3333333333333333) / v) - (single(4.0) * (u / 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 < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 59.3%

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

      1. associate-*r/ 59.3%

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

      2. metadata-eval 59.3%

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

   9. Simplified 59.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(\frac{1.3333333333333333}{v} - 4 \cdot \frac{u}{v}\right) + u \cdot \left(2 - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(u \cdot \left(2 + \frac{4}{v}\right) - \left(2 + \frac{1.3333333333333333}{v}\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (-
     (* u 2.0)
     (/
      (* u (- (* u (+ 2.0 (/ 4.0 v))) (+ 2.0 (/ 1.3333333333333333 v))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - ((u * ((u * (2.0f + (4.0f / v))) - (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - ((u * ((u * (2.0e0 + (4.0e0 / v))) - (2.0e0 + (1.3333333333333333e0 / v)))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(u * Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(4.0) / v))) - 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(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - ((u * ((u * (single(2.0) + (single(4.0) / v))) - (single(2.0) + (single(1.3333333333333333) / v)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 59.3%

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

      1. associate-*r/ 59.3%

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

      2. metadata-eval 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in u around 0 59.3%

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

       1. associate-*r/ 59.3%

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

       2. metadata-eval 59.3%

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

       3. associate-*r/ 59.3%

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

       4. metadata-eval 59.3%

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

   12. Simplified 59.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(u \cdot \left(2 + \frac{4}{v}\right) - \left(2 + \frac{1.3333333333333333}{v}\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(u \cdot 2 - 2\right) + u \cdot \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (-
     (* u 2.0)
     (/ (+ (* u (- (* u 2.0) 2.0)) (* u (/ -1.3333333333333333 v))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - (((u * ((u * 2.0f) - 2.0f)) + (u * (-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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - (((u * ((u * 2.0e0) - 2.0e0)) + (u * ((-1.3333333333333333e0) / v))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(u * Float32(Float32(u * Float32(2.0)) - Float32(2.0))) + Float32(u * Float32(Float32(-1.3333333333333333) / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - (((u * ((u * single(2.0)) - single(2.0))) + (u * (single(-1.3333333333333333) / v))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 59.3%

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

      1. associate-*r/ 59.3%

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

      2. metadata-eval 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in u around 0 57.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(u \cdot 2 - 2\right) + u \cdot \frac{-1.3333333333333333}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(2 - u \cdot 2\right) - \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (+
     (* u 2.0)
     (/ (- (* u (- 2.0 (* u 2.0))) (* (/ u v) -1.3333333333333333)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) + (((u * (2.0f - (u * 2.0f))) - ((u / v) * -1.3333333333333333f)) / v));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) + (((u * (2.0e0 - (u * 2.0e0))) - ((u / v) * (-1.3333333333333333e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) + Float32(Float32(Float32(u * Float32(Float32(2.0) - Float32(u * Float32(2.0)))) - Float32(Float32(u / v) * Float32(-1.3333333333333333))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) + (((u * (single(2.0) - (u * single(2.0)))) - ((u / v) * single(-1.3333333333333333))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 57.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 + \frac{u \cdot \left(2 - u \cdot 2\right) - \frac{u}{v} \cdot -1.3333333333333333}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(u \cdot 2 - \frac{\frac{u}{v} \cdot -1.3333333333333333 + u \cdot -2}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    1.0
    (-
     (- (* u 2.0) (/ (+ (* (/ u v) -1.3333333333333333) (* u -2.0)) v))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((u * 2.0f) - ((((u / v) * -1.3333333333333333f) + (u * -2.0f)) / v)) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((u * 2.0e0) - ((((u / v) * (-1.3333333333333333e0)) + (u * (-2.0e0))) / v)) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(u / v) * Float32(-1.3333333333333333)) + Float32(u * Float32(-2.0))) / v)) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((u * single(2.0)) - ((((u / v) * single(-1.3333333333333333)) + (u * single(-2.0))) / v)) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in u around 0 52.4%

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

   4. Taylor expanded in v around -inf 54.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(u \cdot 2 - \frac{\frac{u}{v} \cdot -1.3333333333333333 + u \cdot -2}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{\frac{u}{v} \cdot -1.3333333333333333 + u \cdot -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+
    -1.0
    (- (* u 2.0) (/ (+ (* (/ u v) -1.3333333333333333) (* u -2.0)) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + ((u * 2.0f) - ((((u / v) * -1.3333333333333333f) + (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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + ((u * 2.0e0) - ((((u / v) * (-1.3333333333333333e0)) + (u * (-2.0e0))) / v))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(u / v) * Float32(-1.3333333333333333)) + Float32(u * Float32(-2.0))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + ((u * single(2.0)) - ((((u / v) * single(-1.3333333333333333)) + (u * single(-2.0))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

   3. Taylor expanded in u around 0 52.4%

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

   4. Taylor expanded in v around -inf 54.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{\frac{u}{v} \cdot -1.3333333333333333 + u \cdot -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.0% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(-2 + \frac{-1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   1.0
   (+ -1.0 (- (* u 2.0) (/ (* u (+ -2.0 (/ -1.3333333333333333 v))) v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f;
	} 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 <= 0.10000000149011612e0) then
        tmp = 1.0e0
    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(0.10000000149011612))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(u * Float32(2.0)) - Float32(Float32(u * 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(0.10000000149011612))
		tmp = single(1.0);
	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 < 0.100000001

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-undefine 100.0%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. fma-define 100.0%

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

      9. pow2 100.0%

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

      10. +-commutative 100.0%

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

      11. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 93.9%

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

   if 0.100000001 < v

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. fma-define 95.5%

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

      3. +-commutative 95.5%

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

      4. fma-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in u around 0 59.1%

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

   6. Taylor expanded in v around -inf 59.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{u \cdot \left(1.3333333333333333 + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)\right)}{v} + u \cdot \left(2 \cdot u - 2\right)}{v} + 2 \cdot u\right)} - 1 \]

   7. Taylor expanded in u around 0 59.3%

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

      1. associate-*r/ 59.3%

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

      2. metadata-eval 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in u around 0 54.5%

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

       1. mul-1-neg 54.5%

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

       2. distribute-rgt-neg-in 54.5%

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

       3. associate-*r/ 54.5%

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

       4. metadata-eval 54.5%

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

       5. distribute-neg-in 54.5%

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

       6. metadata-eval 54.5%

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

       7. distribute-neg-frac 54.5%

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

       8. metadata-eval 54.5%

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

   12. Simplified 54.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(u \cdot 2 - \frac{u \cdot \left(-2 + \frac{-1.3333333333333333}{v}\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f - (u * 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 <= 0.25e0) then
        tmp = 1.0e0
    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(0.25))
		tmp = Float32(1.0);
	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(0.25))
		tmp = single(1.0);
	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 < 0.25

   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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-undefine 99.9%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 99.9%

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

      8. fma-define 99.9%

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

      9. pow2 99.9%

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

      10. +-commutative 99.9%

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

      11. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.5%

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

   if 0.25 < 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 94.4%

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

      3. +-commutative 94.4%

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

      4. fma-define 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 78.3%

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

   6. Taylor expanded in v around -inf 61.3%

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

4. Final simplification 90.7%

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

Alternative 16: 90.6% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-undefine 99.9%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*l* 99.9%

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

      8. fma-define 99.9%

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

      9. pow2 99.9%

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

      10. +-commutative 99.9%

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

      11. fma-undefine 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 92.5%

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

   if 0.25 < v

   1. Initial program 93.7%

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

   3. Taylor expanded in u around 0 65.9%

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

   4. Taylor expanded in v around inf 60.7%

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

      1. sub-neg 60.7%

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

      2. distribute-lft-out 60.7%

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

      3. metadata-eval 60.7%

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

   6. Simplified 60.7%

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

4. Final simplification 90.7%

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

Alternative 17: 86.5% 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 99.6%

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

   1. +-commutative 99.6%

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

   2. +-commutative 99.6%

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

   3. fma-undefine 99.5%

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

   4. add-cube-cbrt 99.4%

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

   5. fma-undefine 99.4%

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

   6. +-commutative 99.4%

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

   7. associate-*l* 99.4%

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

   8. fma-define 99.5%

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

   9. pow2 99.5%

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

   10. +-commutative 99.5%

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

   11. fma-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in v around 0 87.6%

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

Alternative 18: 6.2% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in u around 0 5.4%

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

Reproduce

herbie shell --seed 2024144 
(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))))))))