HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 11.8s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ t_1 := 4 \cdot t\_0\\ t_2 := e^{\frac{-2}{v}}\\ \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(\frac{u \cdot 2 - \frac{\frac{0.5 \cdot \left(\left(u \cdot 32 - t\_1\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(t\_1 - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot t\_0\right)\right) - u \cdot 8}{v}}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v} + v \cdot \left(\frac{-1}{t\_2} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0)))
        (t_1 (* 4.0 t_0))
        (t_2 (exp (/ -2.0 v))))
   (if (<= v 0.4000000059604645)
     (+ 1.0 (* v (log (+ u (* u t_2)))))
     (-
      -1.0
      (*
       u
       (+
        (/
         (-
          (* u 2.0)
          (/
           (+
            (/
             (+
              (* 0.5 (- (- (* u 32.0) t_1) (* u 9.333333333333334)))
              (*
               -0.5
               (/
                (-
                 (+
                  (* 4.0 (+ (* u 9.333333333333334) (- t_1 (* u 32.0))))
                  (- (* u 42.666666666666664) (* 8.0 t_0)))
                 (* u 8.0))
                v)))
             v)
            (* (- (* u 8.0) (* u 16.0)) 0.5))
           v))
         v)
        (* v (- (/ -1.0 t_2) -1.0))))))))

C

float code(float u, float v) {
	float t_0 = (u * 16.0f) - (u * 8.0f);
	float t_1 = 4.0f * t_0;
	float t_2 = expf((-2.0f / v));
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((u + (u * t_2))));
	} else {
		tmp = -1.0f - (u * ((((u * 2.0f) - (((((0.5f * (((u * 32.0f) - t_1) - (u * 9.333333333333334f))) + (-0.5f * ((((4.0f * ((u * 9.333333333333334f) + (t_1 - (u * 32.0f)))) + ((u * 42.666666666666664f) - (8.0f * t_0))) - (u * 8.0f)) / v))) / v) + (((u * 8.0f) - (u * 16.0f)) * 0.5f)) / v)) / v) + (v * ((-1.0f / t_2) - -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: t_2
    real(4) :: tmp
    t_0 = (u * 16.0e0) - (u * 8.0e0)
    t_1 = 4.0e0 * t_0
    t_2 = exp(((-2.0e0) / v))
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log((u + (u * t_2))))
    else
        tmp = (-1.0e0) - (u * ((((u * 2.0e0) - (((((0.5e0 * (((u * 32.0e0) - t_1) - (u * 9.333333333333334e0))) + ((-0.5e0) * ((((4.0e0 * ((u * 9.333333333333334e0) + (t_1 - (u * 32.0e0)))) + ((u * 42.666666666666664e0) - (8.0e0 * t_0))) - (u * 8.0e0)) / v))) / v) + (((u * 8.0e0) - (u * 16.0e0)) * 0.5e0)) / v)) / v) + (v * (((-1.0e0) / t_2) - (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))
	t_1 = Float32(Float32(4.0) * t_0)
	t_2 = exp(Float32(Float32(-2.0) / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(u * t_2)))));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(Float32(u * Float32(32.0)) - t_1) - Float32(u * Float32(9.333333333333334)))) + Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(9.333333333333334)) + Float32(t_1 - Float32(u * Float32(32.0))))) + Float32(Float32(u * Float32(42.666666666666664)) - Float32(Float32(8.0) * t_0))) - Float32(u * Float32(8.0))) / v))) / v) + Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5))) / v)) / v) + Float32(v * Float32(Float32(Float32(-1.0) / t_2) - Float32(-1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (u * single(16.0)) - (u * single(8.0));
	t_1 = single(4.0) * t_0;
	t_2 = exp((single(-2.0) / v));
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log((u + (u * t_2))));
	else
		tmp = single(-1.0) - (u * ((((u * single(2.0)) - (((((single(0.5) * (((u * single(32.0)) - t_1) - (u * single(9.333333333333334)))) + (single(-0.5) * ((((single(4.0) * ((u * single(9.333333333333334)) + (t_1 - (u * single(32.0))))) + ((u * single(42.666666666666664)) - (single(8.0) * t_0))) - (u * single(8.0))) / v))) / v) + (((u * single(8.0)) - (u * single(16.0))) * single(0.5))) / v)) / v) + (v * ((single(-1.0) / t_2) - single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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 u around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.6%

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

      5. sqr-neg 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. fma-undefine 99.6%

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

      2. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   if 0.400000006 < 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 89.4%

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

   4. Taylor expanded in v around -inf 87.4%

      \[\leadsto u \cdot \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.5 \cdot \frac{8 \cdot u - \left(4 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + \left(8 \cdot \left(8 \cdot u - 16 \cdot u\right) + 42.666666666666664 \cdot u\right)\right)}{v} + 0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right)}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u \cdot e^{\frac{-2}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(\frac{u \cdot 2 - \frac{\frac{0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)\right) - u \cdot 8}{v}}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v} + v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ t_1 := 4 \cdot t\_0\\ \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(\frac{u \cdot 2 - \frac{\frac{0.5 \cdot \left(\left(u \cdot 32 - t\_1\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(t\_1 - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot t\_0\right)\right) - u \cdot 8}{v}}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v} + v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))) (t_1 (* 4.0 t_0)))
   (if (<= v 0.4000000059604645)
     (+ 1.0 (* v (log (* u (- (expm1 (/ -2.0 v)))))))
     (-
      -1.0
      (*
       u
       (+
        (/
         (-
          (* u 2.0)
          (/
           (+
            (/
             (+
              (* 0.5 (- (- (* u 32.0) t_1) (* u 9.333333333333334)))
              (*
               -0.5
               (/
                (-
                 (+
                  (* 4.0 (+ (* u 9.333333333333334) (- t_1 (* u 32.0))))
                  (- (* u 42.666666666666664) (* 8.0 t_0)))
                 (* u 8.0))
                v)))
             v)
            (* (- (* u 8.0) (* u 16.0)) 0.5))
           v))
         v)
        (* v (- (/ -1.0 (exp (/ -2.0 v))) -1.0))))))))

C

float code(float u, float v) {
	float t_0 = (u * 16.0f) - (u * 8.0f);
	float t_1 = 4.0f * t_0;
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((u * -expm1f((-2.0f / v)))));
	} else {
		tmp = -1.0f - (u * ((((u * 2.0f) - (((((0.5f * (((u * 32.0f) - t_1) - (u * 9.333333333333334f))) + (-0.5f * ((((4.0f * ((u * 9.333333333333334f) + (t_1 - (u * 32.0f)))) + ((u * 42.666666666666664f) - (8.0f * t_0))) - (u * 8.0f)) / v))) / v) + (((u * 8.0f) - (u * 16.0f)) * 0.5f)) / v)) / v) + (v * ((-1.0f / expf((-2.0f / v))) - -1.0f))));
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))
	t_1 = Float32(Float32(4.0) * t_0)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(-expm1(Float32(Float32(-2.0) / v)))))));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(Float32(u * Float32(32.0)) - t_1) - Float32(u * Float32(9.333333333333334)))) + Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(9.333333333333334)) + Float32(t_1 - Float32(u * Float32(32.0))))) + Float32(Float32(u * Float32(42.666666666666664)) - Float32(Float32(8.0) * t_0))) - Float32(u * Float32(8.0))) / v))) / v) + Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5))) / v)) / v) + Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - Float32(-1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 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 u around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in u around -inf 99.6%

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

      1. expm1-define 99.6%

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

      2. associate-*r* 99.6%

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

      3. mul-1-neg 99.6%

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

   8. Simplified 99.6%

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

   if 0.400000006 < 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 89.4%

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

   4. Taylor expanded in v around -inf 87.4%

      \[\leadsto u \cdot \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.5 \cdot \frac{8 \cdot u - \left(4 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + \left(8 \cdot \left(8 \cdot u - 16 \cdot u\right) + 42.666666666666664 \cdot u\right)\right)}{v} + 0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right)}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(\frac{u \cdot 2 - \frac{\frac{0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)\right) - u \cdot 8}{v}}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v} + v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ t_1 := 4 \cdot t\_0\\ \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(\frac{u \cdot 2 - \frac{\frac{0.5 \cdot \left(\left(u \cdot 32 - t\_1\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(t\_1 - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot t\_0\right)\right) - u \cdot 8}{v}}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v} + v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))) (t_1 (* 4.0 t_0)))
   (if (<= v 0.4000000059604645)
     (+ 1.0 (* v (log (+ u u))))
     (-
      -1.0
      (*
       u
       (+
        (/
         (-
          (* u 2.0)
          (/
           (+
            (/
             (+
              (* 0.5 (- (- (* u 32.0) t_1) (* u 9.333333333333334)))
              (*
               -0.5
               (/
                (-
                 (+
                  (* 4.0 (+ (* u 9.333333333333334) (- t_1 (* u 32.0))))
                  (- (* u 42.666666666666664) (* 8.0 t_0)))
                 (* u 8.0))
                v)))
             v)
            (* (- (* u 8.0) (* u 16.0)) 0.5))
           v))
         v)
        (* v (- (/ -1.0 (exp (/ -2.0 v))) -1.0))))))))

C

float code(float u, float v) {
	float t_0 = (u * 16.0f) - (u * 8.0f);
	float t_1 = 4.0f * t_0;
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((u + u)));
	} else {
		tmp = -1.0f - (u * ((((u * 2.0f) - (((((0.5f * (((u * 32.0f) - t_1) - (u * 9.333333333333334f))) + (-0.5f * ((((4.0f * ((u * 9.333333333333334f) + (t_1 - (u * 32.0f)))) + ((u * 42.666666666666664f) - (8.0f * t_0))) - (u * 8.0f)) / v))) / v) + (((u * 8.0f) - (u * 16.0f)) * 0.5f)) / v)) / v) + (v * ((-1.0f / expf((-2.0f / v))) - -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (u * 16.0e0) - (u * 8.0e0)
    t_1 = 4.0e0 * t_0
    if (v <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log((u + u)))
    else
        tmp = (-1.0e0) - (u * ((((u * 2.0e0) - (((((0.5e0 * (((u * 32.0e0) - t_1) - (u * 9.333333333333334e0))) + ((-0.5e0) * ((((4.0e0 * ((u * 9.333333333333334e0) + (t_1 - (u * 32.0e0)))) + ((u * 42.666666666666664e0) - (8.0e0 * t_0))) - (u * 8.0e0)) / v))) / v) + (((u * 8.0e0) - (u * 16.0e0)) * 0.5e0)) / v)) / v) + (v * (((-1.0e0) / exp(((-2.0e0) / v))) - (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))
	t_1 = Float32(Float32(4.0) * t_0)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + u))));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(Float32(Float32(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(Float32(u * Float32(32.0)) - t_1) - Float32(u * Float32(9.333333333333334)))) + Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(9.333333333333334)) + Float32(t_1 - Float32(u * Float32(32.0))))) + Float32(Float32(u * Float32(42.666666666666664)) - Float32(Float32(8.0) * t_0))) - Float32(u * Float32(8.0))) / v))) / v) + Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5))) / v)) / v) + Float32(v * Float32(Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))) - Float32(-1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (u * single(16.0)) - (u * single(8.0));
	t_1 = single(4.0) * t_0;
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log((u + u)));
	else
		tmp = single(-1.0) - (u * ((((u * single(2.0)) - (((((single(0.5) * (((u * single(32.0)) - t_1) - (u * single(9.333333333333334)))) + (single(-0.5) * ((((single(4.0) * ((u * single(9.333333333333334)) + (t_1 - (u * single(32.0))))) + ((u * single(42.666666666666664)) - (single(8.0) * t_0))) - (u * single(8.0))) / v))) / v) + (((u * single(8.0)) - (u * single(16.0))) * single(0.5))) / v)) / v) + (v * ((single(-1.0) / exp((single(-2.0) / v))) - single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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 u around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.6%

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

      5. sqr-neg 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. fma-undefine 99.6%

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

      2. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in v around inf 94.2%

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

   if 0.400000006 < 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 89.4%

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

   4. Taylor expanded in v around -inf 87.4%

      \[\leadsto u \cdot \left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.5 \cdot \frac{8 \cdot u - \left(4 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right) + \left(8 \cdot \left(8 \cdot u - 16 \cdot u\right) + 42.666666666666664 \cdot u\right)\right)}{v} + 0.5 \cdot \left(9.333333333333334 \cdot u - \left(4 \cdot \left(8 \cdot u - 16 \cdot u\right) + 32 \cdot u\right)\right)}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v}} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(\frac{u \cdot 2 - \frac{\frac{0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)\right) - u \cdot 8}{v}}{v} + \left(u \cdot 8 - u \cdot 16\right) \cdot 0.5}{v}}{v} + v \cdot \left(\frac{-1}{e^{\frac{-2}{v}}} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 - \frac{-1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 - \frac{\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 - -0.5 \cdot \frac{\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334}{v}}{v}}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log (+ u u))))
   (+
    -1.0
    (*
     u
     (-
      (* v (- -1.0 (/ -1.0 (exp (/ -2.0 v)))))
      (/
       (-
        (* u 2.0)
        (/
         (-
          (* (- (* u 8.0) (* u 16.0)) 0.5)
          (*
           -0.5
           (/
            (-
             (- (* u 32.0) (* 4.0 (- (* u 16.0) (* u 8.0))))
             (* u 9.333333333333334))
            v)))
         v))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((u + u)));
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f - (-1.0f / expf((-2.0f / v))))) - (((u * 2.0f) - (((((u * 8.0f) - (u * 16.0f)) * 0.5f) - (-0.5f * ((((u * 32.0f) - (4.0f * ((u * 16.0f) - (u * 8.0f)))) - (u * 9.333333333333334f)) / 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log((u + u)))
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) - ((-1.0e0) / exp(((-2.0e0) / v))))) - (((u * 2.0e0) - (((((u * 8.0e0) - (u * 16.0e0)) * 0.5e0) - ((-0.5e0) * ((((u * 32.0e0) - (4.0e0 * ((u * 16.0e0) - (u * 8.0e0)))) - (u * 9.333333333333334e0)) / v))) / v)) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + 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(Float32(u * Float32(2.0)) - Float32(Float32(Float32(Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))) * Float32(0.5)) - Float32(Float32(-0.5) * Float32(Float32(Float32(Float32(u * Float32(32.0)) - Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))))) - Float32(u * Float32(9.333333333333334))) / v))) / v)) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0) + (v * log((u + u)));
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) - (single(-1.0) / exp((single(-2.0) / v))))) - (((u * single(2.0)) - (((((u * single(8.0)) - (u * single(16.0))) * single(0.5)) - (single(-0.5) * ((((u * single(32.0)) - (single(4.0) * ((u * single(16.0)) - (u * single(8.0))))) - (u * single(9.333333333333334))) / v))) / v)) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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 u around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.6%

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

      5. sqr-neg 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. fma-undefine 99.6%

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

      2. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in v around inf 94.2%

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

   if 0.400000006 < 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 89.4%

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

   4. Taylor expanded in v around -inf 86.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 - \frac{-1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 - \frac{\left(u \cdot 8 - u \cdot 16\right) \cdot 0.5 - -0.5 \cdot \frac{\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334}{v}}{v}}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v} - v \cdot \left(-1 - \frac{-1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   (+ 1.0 (* v (log (+ u u))))
   (-
    -1.0
    (*
     u
     (-
      (* -0.5 (/ (+ (* u -4.0) (* -8.0 (/ u v))) v))
      (* v (- -1.0 (/ -1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f + (v * logf((u + u)));
	} else {
		tmp = -1.0f - (u * ((-0.5f * (((u * -4.0f) + (-8.0f * (u / v))) / 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (v * log((u + u)))
    else
        tmp = (-1.0e0) - (u * (((-0.5e0) * (((u * (-4.0e0)) + ((-8.0e0) * (u / v))) / 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(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + u))));
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(-8.0) * Float32(u / v))) / 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(0.4000000059604645))
		tmp = single(1.0) + (v * log((u + u)));
	else
		tmp = single(-1.0) - (u * ((single(-0.5) * (((u * single(-4.0)) + (single(-8.0) * (u / v))) / 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 < 0.400000006

   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 u around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.6%

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

      5. sqr-neg 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. fma-undefine 99.6%

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

      2. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in v around inf 94.2%

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

   if 0.400000006 < 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 89.4%

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

   4. Taylor expanded in v around -inf 85.0%

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

      1. mul-1-neg 85.0%

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

      2. distribute-neg-frac2 85.0%

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

      3. associate--l+ 85.0%

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

      4. *-commutative 85.0%

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

      5. distribute-rgt-out-- 85.0%

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

      6. metadata-eval 85.0%

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

      7. *-commutative 85.0%

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

   6. Simplified 85.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + -8 \cdot \frac{u}{v}}{v} - v \cdot \left(-1 - \frac{-1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   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 u around inf 99.6%

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

      1. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. fma-define 99.6%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.6%

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

      5. sqr-neg 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. fma-undefine 99.6%

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

      2. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in v around inf 94.2%

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

   if 0.400000006 < 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 89.4%

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

   4. Taylor expanded in v around inf 82.3%

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

      1. sub-neg 82.3%

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

      2. rec-exp 82.3%

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

      3. metadata-eval 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. metadata-eval 82.3%

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

      2. sub-neg 82.3%

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

      3. expm1-undefine 82.3%

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

      4. distribute-neg-frac 82.3%

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

      5. metadata-eval 82.3%

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

   8. Simplified 82.3%

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

4. Final simplification 93.4%

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

Alternative 8: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 + v \cdot \log \left(u + u\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 - \left(0.5 \cdot t\_0 - \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot t\_0 - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))))
   (if (<= v 0.20000000298023224)
     (+ 1.0 (* v (log (+ u u))))
     (+
      -1.0
      (*
       u
       (+
        2.0
        (/
         (+
          2.0
          (-
           (/
            (-
             1.3333333333333333
             (-
              (* 0.5 t_0)
              (/
               (-
                0.6666666666666666
                (* (+ (* u 9.333333333333334) (- (* 4.0 t_0) (* u 32.0))) 0.5))
               v)))
            v)
           (* u 2.0)))
         v)))))))

C

float code(float u, float v) {
	float t_0 = (u * 16.0f) - (u * 8.0f);
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f + (v * logf((u + u)));
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f - ((0.5f * t_0) - ((0.6666666666666666f - (((u * 9.333333333333334f) + ((4.0f * t_0) - (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) :: t_0
    real(4) :: tmp
    t_0 = (u * 16.0e0) - (u * 8.0e0)
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0 + (v * log((u + u)))
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 - ((0.5e0 * t_0) - ((0.6666666666666666e0 - (((u * 9.333333333333334e0) + ((4.0e0 * t_0) - (u * 32.0e0))) * 0.5e0)) / v))) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u + 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(0.5) * t_0) - Float32(Float32(Float32(0.6666666666666666) - Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * t_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)
	t_0 = (u * single(16.0)) - (u * single(8.0));
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) + (v * log((u + u)));
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) - ((single(0.5) * t_0) - ((single(0.6666666666666666) - (((u * single(9.333333333333334)) + ((single(4.0) * t_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 u around inf 99.8%

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

      1. neg-mul-1 99.8%

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

   5. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.8%

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

      5. sqr-neg 99.8%

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

      6. sqrt-unprod 99.8%

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

      7. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in v around inf 94.5%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around -inf 78.4%

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

4. Final simplification 93.4%

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

Alternative 9: 91.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 - \left(0.5 \cdot t\_0 - \frac{0.6666666666666666 - \left(u \cdot 9.333333333333334 + \left(4 \cdot t\_0 - u \cdot 32\right)\right) \cdot 0.5}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))))
   (if (<= v 0.20000000298023224)
     1.0
     (+
      -1.0
      (*
       u
       (+
        2.0
        (/
         (+
          2.0
          (-
           (/
            (-
             1.3333333333333333
             (-
              (* 0.5 t_0)
              (/
               (-
                0.6666666666666666
                (* (+ (* u 9.333333333333334) (- (* 4.0 t_0) (* u 32.0))) 0.5))
               v)))
            v)
           (* u 2.0)))
         v)))))))

C

float code(float u, float v) {
	float t_0 = (u * 16.0f) - (u * 8.0f);
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f - ((0.5f * t_0) - ((0.6666666666666666f - (((u * 9.333333333333334f) + ((4.0f * t_0) - (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) :: t_0
    real(4) :: tmp
    t_0 = (u * 16.0e0) - (u * 8.0e0)
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 - ((0.5e0 * t_0) - ((0.6666666666666666e0 - (((u * 9.333333333333334e0) + ((4.0e0 * t_0) - (u * 32.0e0))) * 0.5e0)) / v))) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		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(1.3333333333333333) - Float32(Float32(Float32(0.5) * t_0) - Float32(Float32(Float32(0.6666666666666666) - Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * t_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)
	t_0 = (u * single(16.0)) - (u * single(8.0));
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) - ((single(0.5) * t_0) - ((single(0.6666666666666666) - (((u * single(9.333333333333334)) + ((single(4.0) * t_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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around -inf 78.4%

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

4. Final simplification 91.9%

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

Alternative 10: 91.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((-2.0f * (u / v)) + (v * (-1.0f + (1.0f - ((-2.0f + ((-2.0f + (-1.3333333333333333f / 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
    else
        tmp = (-1.0e0) + (u * (((-2.0e0) * (u / v)) + (v * ((-1.0e0) + (1.0e0 - (((-2.0e0) + (((-2.0e0) + ((-1.3333333333333333e0) / 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(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-2.0) * Float32(u / v)) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-2.0) + Float32(Float32(-1.3333333333333333) / 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);
	else
		tmp = single(-1.0) + (u * ((single(-2.0) * (u / v)) + (v * (single(-1.0) + (single(1.0) - ((single(-2.0) + ((single(-2.0) + (single(-1.3333333333333333) / 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. 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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around inf 77.9%

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

   5. Taylor expanded in v around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. unsub-neg 74.7%

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

      3. sub-neg 74.7%

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

      4. associate-*r/ 74.7%

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

      5. distribute-lft-in 74.7%

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

      6. metadata-eval 74.7%

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

      7. neg-mul-1 74.7%

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

      8. associate-*r/ 74.7%

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

      9. metadata-eval 74.7%

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

      10. distribute-neg-frac 74.7%

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

      11. metadata-eval 74.7%

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

      12. metadata-eval 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 91.6%

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

Alternative 11: 91.4% accurate, 7.1× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (* -2.0 (/ u v))
      (+
       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;
	} else {
		tmp = -1.0f + (u * ((-2.0f * (u / v)) + (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
    else
        tmp = (-1.0e0) + (u * (((-2.0e0) * (u / v)) + (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(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(-2.0) * Float32(u / v)) + 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);
	else
		tmp = single(-1.0) + (u * ((single(-2.0) * (u / v)) + (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. 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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around inf 77.9%

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

   5. Taylor expanded in v around -inf 75.9%

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

4. Final simplification 91.7%

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

Alternative 12: 91.2% accurate, 8.2× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   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 <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around inf 77.9%

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

   5. Taylor expanded in v around -inf 74.7%

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

4. Final simplification 91.6%

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

Alternative 13: 91.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (v * ((2.0f + ((2.0f + (1.3333333333333333f / 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
    else
        tmp = 1.0e0 + ((u * (v * ((2.0e0 + ((2.0e0 + (1.3333333333333333e0 / 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(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(v * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / 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);
	else
		tmp = single(1.0) + ((u * (v * ((single(2.0) + ((single(2.0) + (single(1.3333333333333333) / 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. 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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 74.7%

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

   4. Taylor expanded in v around -inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. distribute-neg-frac2 72.3%

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

      3. sub-neg 72.3%

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

      4. associate-*r/ 72.3%

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

      5. distribute-lft-in 72.3%

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

      6. metadata-eval 72.3%

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

      7. neg-mul-1 72.3%

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

      8. associate-*r/ 72.3%

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

      9. metadata-eval 72.3%

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

      10. distribute-neg-frac 72.3%

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

      11. metadata-eval 72.3%

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

      12. metadata-eval 72.3%

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

   6. Simplified 72.3%

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

      1. associate-*r/ 72.3%

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

      2. frac-2neg 72.3%

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

      3. +-commutative 72.3%

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

      4. remove-double-neg 72.3%

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

   8. Applied egg-rr 72.3%

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

      1. distribute-frac-neg 72.3%

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

      2. associate-*r/ 72.3%

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

      3. distribute-rgt-neg-in 72.3%

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

      4. distribute-neg-frac 72.3%

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

      5. distribute-neg-in 72.3%

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

      6. metadata-eval 72.3%

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

      7. distribute-neg-frac 72.3%

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

      8. distribute-neg-in 72.3%

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

      9. metadata-eval 72.3%

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

      10. distribute-neg-frac 72.3%

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

      11. metadata-eval 72.3%

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

   10. Simplified 72.3%

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

Alternative 14: 91.0% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   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 <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 74.7%

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

   4. Taylor expanded in v around -inf 72.4%

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

4. Final simplification 91.5%

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

Alternative 15: 90.9% accurate, 11.8× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around inf 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/l* 67.4%

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

      3. distribute-lft-out 67.4%

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

   6. Simplified 67.4%

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

4. Final simplification 91.1%

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

Alternative 16: 90.8% accurate, 13.3× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f - (2.0f - (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
    else
        tmp = 1.0e0 - (2.0e0 - (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(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(2.0) - 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);
	else
		tmp = single(1.0) - (single(2.0) - (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. 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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 74.7%

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

   4. Taylor expanded in v around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. metadata-eval 66.9%

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

      3. metadata-eval 66.9%

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

      4. distribute-neg-frac 66.9%

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

      5. unsub-neg 66.9%

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

   6. Simplified 66.9%

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

4. Final simplification 91.1%

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

Alternative 17: 90.8% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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 < 0.200000003

   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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 74.7%

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

   4. Taylor expanded in v around inf 66.8%

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

      1. sub-neg 66.8%

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

      2. distribute-lft-out 66.8%

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

      3. metadata-eval 66.8%

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

   6. Simplified 66.8%

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

4. Final simplification 91.1%

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

Alternative 18: 90.2% accurate, 17.7× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} 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
    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(1.0);
	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);
	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. 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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around inf 57.7%

      \[\leadsto u \cdot \color{blue}{2} - 1 \]

   5. Taylor expanded in u around inf 57.7%

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

4. Final simplification 90.4%

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

Alternative 19: 90.2% accurate, 21.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   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. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.3%

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

   3. Taylor expanded in u around 0 85.5%

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

   4. Taylor expanded in v around inf 57.7%

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

4. Final simplification 90.4%

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

Alternative 20: 87.1% 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 99.5%

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

   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. Taylor expanded in v around 0 86.7%

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

Alternative 21: 5.9% 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. Taylor expanded in u around 0 6.3%

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

Reproduce

herbie shell --seed 2024131 
(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))))))))