HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.9s

Alternatives: 23

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.3%

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

3. Simplified 99.3%

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

   1. fma-undefine 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	float t_1 = (u * 16.0f) - (u * 8.0f);
	float t_2 = 4.0f * t_1;
	float tmp;
	if (v <= 0.5f) {
		tmp = 1.0f + (v * logf((u + t_0)));
	} else {
		tmp = 1.0f - (2.0f - (u * ((v * (-1.0f + (1.0f / t_0))) - (((((0.5f * t_1) - (((0.5f * (((u * 32.0f) - t_2) - (u * 9.333333333333334f))) + (-0.5f * ((((4.0f * ((u * 9.333333333333334f) + (t_2 - (u * 32.0f)))) + ((u * 42.666666666666664f) - (8.0f * t_1))) - (u * 8.0f)) / v))) / 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) :: t_1
    real(4) :: t_2
    real(4) :: tmp
    t_0 = exp(((-2.0e0) / v))
    t_1 = (u * 16.0e0) - (u * 8.0e0)
    t_2 = 4.0e0 * t_1
    if (v <= 0.5e0) then
        tmp = 1.0e0 + (v * log((u + t_0)))
    else
        tmp = 1.0e0 - (2.0e0 - (u * ((v * ((-1.0e0) + (1.0e0 / t_0))) - (((((0.5e0 * t_1) - (((0.5e0 * (((u * 32.0e0) - t_2) - (u * 9.333333333333334e0))) + ((-0.5e0) * ((((4.0e0 * ((u * 9.333333333333334e0) + (t_2 - (u * 32.0e0)))) + ((u * 42.666666666666664e0) - (8.0e0 * t_1))) - (u * 8.0e0)) / v))) / v)) / v) + (u * 2.0e0)) / v))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in u around 0 99.5%

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

   if 0.5 < v

   1. Initial program 93.3%

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

   3. Taylor expanded in u around 0 90.4%

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

   4. Taylor expanded in v around -inf 87.7%

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

4. Final simplification 98.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. add-exp-log 99.4%

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

   2. *-commutative 99.4%

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

   3. log-prod 99.4%

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

   4. add-log-exp 99.4%

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

   5. sub-neg 99.4%

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

   6. log1p-define 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in u around 0 97.7%

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

   1. mul-1-neg 97.7%

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

   2. associate-*r/ 97.7%

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

   3. metadata-eval 97.7%

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

7. Simplified 97.7%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ t_1 := 4 \cdot t\_0\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(\log u, v, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{\frac{0.5 \cdot t\_0 - \frac{0.5 \cdot \left(\left(u \cdot 32 - t\_1\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(t\_1 - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot t\_0\right)\right) - u \cdot 8}{v}}{v}}{v} + u \cdot 2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))) (t_1 (* 4.0 t_0)))
   (if (<= v 0.10000000149011612)
     (fma (log u) v 1.0)
     (-
      1.0
      (-
       2.0
       (*
        u
        (-
         (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
         (/
          (+
           (/
            (-
             (* 0.5 t_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))
            v)
           (* u 2.0))
          v))))))))

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 <= 0.10000000149011612f) {
		tmp = fmaf(logf(u), v, 1.0f);
	} else {
		tmp = 1.0f - (2.0f - (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (((((0.5f * t_0) - (((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)) / v) + (u * 2.0f)) / v))));
	}
	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(0.10000000149011612))
		tmp = fma(log(u), v, Float32(1.0));
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(2.0) - Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) - Float32(Float32(Float32(Float32(Float32(Float32(0.5) * t_0) - 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)) / v) + Float32(u * Float32(2.0))) / v)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

   7. Simplified 99.8%

      \[\leadsto 1 + \color{blue}{v \cdot \log u} \]
   8. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-define 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log u, v, 1\right)} \]

   9. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log u, v, 1\right)} \]

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around -inf 81.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(\log u, v, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{\frac{0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - \frac{0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)\right) - u \cdot 8}{v}}{v}}{v} + u \cdot 2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ t_1 := 4 \cdot t\_0\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{\frac{0.5 \cdot t\_0 - \frac{0.5 \cdot \left(\left(u \cdot 32 - t\_1\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(t\_1 - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot t\_0\right)\right) - u \cdot 8}{v}}{v}}{v} + u \cdot 2}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))) (t_1 (* 4.0 t_0)))
   (if (<= v 0.10000000149011612)
     (- 1.0 (* v (log (/ 1.0 u))))
     (-
      1.0
      (-
       2.0
       (*
        u
        (-
         (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
         (/
          (+
           (/
            (-
             (* 0.5 t_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))
            v)
           (* u 2.0))
          v))))))))

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 <= 0.10000000149011612f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f - (2.0f - (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (((((0.5f * t_0) - (((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)) / 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) :: t_1
    real(4) :: tmp
    t_0 = (u * 16.0e0) - (u * 8.0e0)
    t_1 = 4.0e0 * t_0
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 - (2.0e0 - (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) - (((((0.5e0 * t_0) - (((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)) / 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)))
	t_1 = Float32(Float32(4.0) * t_0)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(2.0) - Float32(u * Float32(Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))) - Float32(Float32(Float32(Float32(Float32(Float32(0.5) * t_0) - 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)) / 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));
	t_1 = single(4.0) * t_0;
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) - (single(2.0) - (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) - (((((single(0.5) * t_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)) / v) + (u * single(2.0))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around -inf 81.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{\frac{0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - \frac{0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right) + -0.5 \cdot \frac{\left(4 \cdot \left(u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)\right) + \left(u \cdot 42.666666666666664 - 8 \cdot \left(u \cdot 16 - u \cdot 8\right)\right)\right) - u \cdot 8}{v}}{v}}{v} + u \cdot 2}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot 16 - u \cdot 8\\ \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 + \frac{0.5 \cdot t\_0 - -0.5 \cdot \frac{u \cdot 9.333333333333334 + \left(4 \cdot t\_0 - u \cdot 32\right)}{v}}{v}}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- (* u 16.0) (* u 8.0))))
   (if (<= v 0.10000000149011612)
     (- 1.0 (* v (log (/ 1.0 u))))
     (-
      1.0
      (-
       2.0
       (*
        u
        (-
         (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
         (/
          (+
           (* u 2.0)
           (/
            (-
             (* 0.5 t_0)
             (*
              -0.5
              (/ (+ (* u 9.333333333333334) (- (* 4.0 t_0) (* u 32.0))) v)))
            v))
          v))))))))

C

float code(float u, float v) {
	float t_0 = (u * 16.0f) - (u * 8.0f);
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f - (2.0f - (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (((u * 2.0f) + (((0.5f * t_0) - (-0.5f * (((u * 9.333333333333334f) + ((4.0f * t_0) - (u * 32.0f))) / v))) / v)) / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (u * 16.0e0) - (u * 8.0e0)
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 - (2.0e0 - (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) - (((u * 2.0e0) + (((0.5e0 * t_0) - ((-0.5e0) * (((u * 9.333333333333334e0) + ((4.0e0 * t_0) - (u * 32.0e0))) / v))) / v)) / 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(0.10000000149011612))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(2.0) - 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(0.5) * t_0) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * t_0) - Float32(u * Float32(32.0)))) / v))) / v)) / 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(0.10000000149011612))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) - (single(2.0) - (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) - (((u * single(2.0)) + (((single(0.5) * t_0) - (single(-0.5) * (((u * single(9.333333333333334)) + ((single(4.0) * t_0) - (u * single(32.0)))) / v))) / v)) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around -inf 80.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 - u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - \frac{u \cdot 2 + \frac{0.5 \cdot \left(u \cdot 16 - u \cdot 8\right) - -0.5 \cdot \frac{u \cdot 9.333333333333334 + \left(4 \cdot \left(u \cdot 16 - u \cdot 8\right) - u \cdot 32\right)}{v}}{v}}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 + u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + \frac{u}{v} \cdot -8}{v} - v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (- 1.0 (* v (log (/ 1.0 u))))
   (-
    1.0
    (+
     2.0
     (*
      u
      (-
       (* -0.5 (/ (+ (* u -4.0) (* (/ u v) -8.0)) v))
       (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f - (2.0f + (u * ((-0.5f * (((u * -4.0f) + ((u / v) * -8.0f)) / v)) - (v * (-1.0f + (1.0f / expf((-2.0f / v))))))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 - (2.0e0 + (u * (((-0.5e0) * (((u * (-4.0e0)) + ((u / v) * (-8.0e0))) / v)) - (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(2.0) + Float32(u * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(u / v) * Float32(-8.0))) / v)) - Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) - (single(2.0) + (u * ((single(-0.5) * (((u * single(-4.0)) + ((u / v) * single(-8.0))) / v)) - (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around -inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. distribute-neg-frac2 79.8%

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

      3. associate--l+ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r/ 79.8%

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

      6. associate-*r/ 79.8%

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

      7. div-sub 79.8%

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

      8. distribute-rgt-out-- 79.8%

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

      9. metadata-eval 79.8%

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

      10. *-commutative 79.8%

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

      11. associate-*r/ 79.8%

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

      12. *-commutative 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(2 + u \cdot \left(-0.5 \cdot \frac{u \cdot -4 + \frac{u}{v} \cdot -8}{v} - v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around inf 76.6%

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

4. Final simplification 97.7%

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

Alternative 10: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (- 1.0 (* v (log (/ 1.0 u))))
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+
         2.0
         (-
          (/
           (+
            1.3333333333333333
            (+
             (* 0.5 (- (* u 8.0) (* u 16.0)))
             (/
              (+
               0.6666666666666666
               (*
                0.5
                (-
                 (- (* u 32.0) (* 4.0 (- (* u 16.0) (* u 8.0))))
                 (* u 9.333333333333334))))
              v)))
           v)
          (* u 2.0)))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f - (v * logf((1.0f / u)));
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + (((1.3333333333333333f + ((0.5f * ((u * 8.0f) - (u * 16.0f))) + ((0.6666666666666666f + (0.5f * (((u * 32.0f) - (4.0f * ((u * 16.0f) - (u * 8.0f)))) - (u * 9.333333333333334f)))) / v))) / v) - (u * 2.0f))) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 - (v * log((1.0e0 / u)))
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + ((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) + ((0.6666666666666666e0 + (0.5e0 * (((u * 32.0e0) - (4.0e0 * ((u * 16.0e0) - (u * 8.0e0)))) - (u * 9.333333333333334e0)))) / v))) / v) - (u * 2.0e0))) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) - Float32(v * log(Float32(Float32(1.0) / u))));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) + Float32(Float32(Float32(0.6666666666666666) + Float32(Float32(0.5) * 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) - Float32(u * Float32(2.0)))) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) - (v * log((single(1.0) / u)));
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) + ((single(0.6666666666666666) + (single(0.5) * (((u * single(32.0)) - (single(4.0) * ((u * single(16.0)) - (u * single(8.0))))) - (u * single(9.333333333333334))))) / v))) / v) - (u * single(2.0)))) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around -inf 75.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 - v \cdot \log \left(\frac{1}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.10000000149011612)
   (+ 1.0 (* v (log u)))
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+
         2.0
         (-
          (/
           (+
            1.3333333333333333
            (+
             (* 0.5 (- (* u 8.0) (* u 16.0)))
             (/
              (+
               0.6666666666666666
               (*
                0.5
                (-
                 (- (* u 32.0) (* 4.0 (- (* u 16.0) (* u 8.0))))
                 (* u 9.333333333333334))))
              v)))
           v)
          (* u 2.0)))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.10000000149011612f) {
		tmp = 1.0f + (v * logf(u));
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + (((1.3333333333333333f + ((0.5f * ((u * 8.0f) - (u * 16.0f))) + ((0.6666666666666666f + (0.5f * (((u * 32.0f) - (4.0f * ((u * 16.0f) - (u * 8.0f)))) - (u * 9.333333333333334f)))) / v))) / v) - (u * 2.0f))) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.10000000149011612e0) then
        tmp = 1.0e0 + (v * log(u))
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + ((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) + ((0.6666666666666666e0 + (0.5e0 * (((u * 32.0e0) - (4.0e0 * ((u * 16.0e0) - (u * 8.0e0)))) - (u * 9.333333333333334e0)))) / v))) / v) - (u * 2.0e0))) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) + Float32(v * log(u)));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) + Float32(Float32(Float32(0.6666666666666666) + Float32(Float32(0.5) * 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) - Float32(u * Float32(2.0)))) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.10000000149011612))
		tmp = single(1.0) + (v * log(u));
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) + ((single(0.6666666666666666) + (single(0.5) * (((u * single(32.0)) - (single(4.0) * ((u * single(16.0)) - (u * single(8.0))))) - (u * single(9.333333333333334))))) / v))) / v) - (u * single(2.0)))) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-log-exp 99.9%

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

      5. sub-neg 99.9%

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

      6. log1p-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in u around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   if 0.100000001 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 85.1%

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

   4. Taylor expanded in v around -inf 75.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.10000000149011612:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503)
   1.0
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+
         2.0
         (-
          (/
           (+
            1.3333333333333333
            (+
             (* 0.5 (- (* u 8.0) (* u 16.0)))
             (/
              (+
               0.6666666666666666
               (*
                0.5
                (-
                 (- (* u 32.0) (* 4.0 (- (* u 16.0) (* u 8.0))))
                 (* u 9.333333333333334))))
              v)))
           v)
          (* u 2.0)))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + (((1.3333333333333333f + ((0.5f * ((u * 8.0f) - (u * 16.0f))) + ((0.6666666666666666f + (0.5f * (((u * 32.0f) - (4.0f * ((u * 16.0f) - (u * 8.0f)))) - (u * 9.333333333333334f)))) / v))) / v) - (u * 2.0f))) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + ((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) + ((0.6666666666666666e0 + (0.5e0 * (((u * 32.0e0) - (4.0e0 * ((u * 16.0e0) - (u * 8.0e0)))) - (u * 9.333333333333334e0)))) / v))) / v) - (u * 2.0e0))) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) + Float32(Float32(Float32(0.6666666666666666) + Float32(Float32(0.5) * 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) - Float32(u * Float32(2.0)))) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) + ((single(0.6666666666666666) + (single(0.5) * (((u * single(32.0)) - (single(4.0) * ((u * single(16.0)) - (u * single(8.0))))) - (u * single(9.333333333333334))))) / v))) / v) - (u * single(2.0)))) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 82.1%

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

   4. Taylor expanded in v around -inf 73.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + 1.3333333333333333}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503)
   1.0
   (+
    1.0
    (-
     (*
      u
      (+
       2.0
       (/
        (+
         2.0
         (-
          (/ (+ (* 0.5 (- (* u 8.0) (* u 16.0))) 1.3333333333333333) v)
          (* u 2.0)))
        v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + ((((0.5f * ((u * 8.0f) - (u * 16.0f))) + 1.3333333333333333f) / v) - (u * 2.0f))) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + ((((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) + 1.3333333333333333e0) / v) - (u * 2.0e0))) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) + Float32(1.3333333333333333)) / v) - Float32(u * Float32(2.0)))) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + ((((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) + single(1.3333333333333333)) / v) - (u * single(2.0)))) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0 82.1%

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

   4. Taylor expanded in v around -inf 69.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \left(\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + 1.3333333333333333}{v} - u \cdot 2\right)}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 - 0.5 \cdot \frac{-4 \cdot \left(-1 + u \cdot \left(2 - u\right)\right) + 4 \cdot \left(u + -1\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503)
   1.0
   (+
    1.0
    (-
     (* (- 1.0 u) -2.0)
     (* 0.5 (/ (+ (* -4.0 (+ -1.0 (* u (- 2.0 u)))) (* 4.0 (+ u -1.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) - (0.5f * (((-4.0f * (-1.0f + (u * (2.0f - u)))) + (4.0f * (u + -1.0f))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) - (0.5e0 * ((((-4.0e0) * ((-1.0e0) + (u * (2.0e0 - u)))) + (4.0e0 * (u + (-1.0e0)))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) - Float32(Float32(0.5) * Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - u)))) + Float32(Float32(4.0) * Float32(u + Float32(-1.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) - (single(0.5) * (((single(-4.0) * (single(-1.0) + (u * (single(2.0) - u)))) + (single(4.0) * (u + single(-1.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 63.5%

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

   6. Taylor expanded in u around 0 63.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 - 0.5 \cdot \frac{-4 \cdot \left(-1 + u \cdot \left(2 - u\right)\right) + 4 \cdot \left(u + -1\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{u \cdot \left(4 + u \cdot -4\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503)
   1.0
   (+ 1.0 (+ (* (- 1.0 u) -2.0) (* 0.5 (/ (* u (+ 4.0 (* u -4.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((1.0f - u) * -2.0f) + (0.5f * ((u * (4.0f + (u * -4.0f))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + (((1.0e0 - u) * (-2.0e0)) + (0.5e0 * ((u * (4.0e0 + (u * (-4.0e0)))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - u) * Float32(-2.0)) + Float32(Float32(0.5) * Float32(Float32(u * Float32(Float32(4.0) + Float32(u * Float32(-4.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(1.0) + (((single(1.0) - u) * single(-2.0)) + (single(0.5) * ((u * (single(4.0) + (u * single(-4.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 63.5%

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

   6. Taylor expanded in u around 0 63.5%

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

      1. *-commutative 63.5%

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

   8. Simplified 63.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 - u\right) \cdot -2 + 0.5 \cdot \frac{u \cdot \left(4 + u \cdot -4\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 90.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \left(2 \cdot \frac{-1}{v} - -2 \cdot \frac{u}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503)
   1.0
   (+ -1.0 (* u (- 2.0 (- (* 2.0 (/ -1.0 v)) (* -2.0 (/ u v))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f - ((2.0f * (-1.0f / v)) - (-2.0f * (u / v)))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 - ((2.0e0 * ((-1.0e0) / v)) - ((-2.0e0) * (u / v)))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(2.0) * Float32(Float32(-1.0) / v)) - Float32(Float32(-2.0) * Float32(u / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) - ((single(2.0) * (single(-1.0) / v)) - (single(-2.0) * (u / v)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 63.5%

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

   6. Taylor expanded in u around 0 63.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 - \left(2 \cdot \frac{-1}{v} - -2 \cdot \frac{u}{v}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 90.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-2 \cdot \left(u + -1\right) - 2 \cdot \frac{u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503)
   1.0
   (- 1.0 (- (* -2.0 (+ u -1.0)) (* 2.0 (/ u v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f - ((-2.0f * (u + -1.0f)) - (2.0f * (u / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 - (((-2.0e0) * (u + (-1.0e0))) - (2.0e0 * (u / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) * Float32(u + Float32(-1.0))) - Float32(Float32(2.0) * Float32(u / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(1.0) - ((single(-2.0) * (u + single(-1.0))) - (single(2.0) * (u / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 63.5%

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

   6. Taylor expanded in u around 0 63.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-2 \cdot \left(u + -1\right) - 2 \cdot \frac{u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 90.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(2 \cdot \frac{-1}{v} - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503) 1.0 (- -1.0 (* u (- (* 2.0 (/ -1.0 v)) 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f - (u * ((2.0f * (-1.0f / v)) - 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) - (u * ((2.0e0 * ((-1.0e0) / v)) - 2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) - Float32(u * Float32(Float32(Float32(2.0) * Float32(Float32(-1.0) / v)) - Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(-1.0) - (u * ((single(2.0) * (single(-1.0) / v)) - single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 63.5%

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

   6. Taylor expanded in u around 0 62.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot \left(2 \cdot \frac{-1}{v} - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 90.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(2 + \frac{2}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503) 1.0 (* u (+ (+ 2.0 (/ 2.0 v)) (/ -1.0 u)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = u * ((2.0f + (2.0f / v)) + (-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 <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = u * ((2.0e0 + (2.0e0 / v)) + ((-1.0e0) / u))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(u * Float32(Float32(Float32(2.0) + Float32(Float32(2.0) / v)) + Float32(Float32(-1.0) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = u * ((single(2.0) + (single(2.0) / v)) + (single(-1.0) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 63.5%

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

   6. Taylor expanded in u around 0 63.0%

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

      1. *-commutative 63.0%

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

   8. Simplified 63.0%

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

   9. Taylor expanded in u around inf 62.6%

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

       1. associate-*r/ 62.6%

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

       2. metadata-eval 62.6%

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

   11. Simplified 62.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(\left(2 + \frac{2}{v}\right) + \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 90.1% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503) 1.0 (+ 1.0 (* (- 1.0 u) -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((1.0f - u) * -2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((1.0e0 - u) * (-2.0e0))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - u) * Float32(-2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((single(1.0) - u) * single(-2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 54.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - u\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 90.1% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.09600000083446503) 1.0 (+ -1.0 (* u 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		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 <= 0.09600000083446503e0) 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(0.09600000083446503))
		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(0.09600000083446503))
		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 < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in v around inf 54.7%

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

   6. Taylor expanded in u around 0 54.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 89.4% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.09600000083446503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v) :precision binary32 (if (<= v 0.09600000083446503) 1.0 -1.0))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.09600000083446503f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.09600000083446503e0) then
        tmp = 1.0e0
    else
        tmp = -1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.09600000083446503))
		tmp = Float32(1.0);
	else
		tmp = Float32(-1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.09600000083446503))
		tmp = single(1.0);
	else
		tmp = single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0960000008

   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%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. log-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in v around 0 92.8%

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

   if 0.0960000008 < v

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. fma-define 93.9%

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

      3. +-commutative 93.9%

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

      4. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in u around 0 46.7%

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

Alternative 23: 6.0% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

   3. +-commutative 99.4%

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

   4. fma-define 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in u around 0 7.2%

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

Reproduce

herbie shell --seed 2024170 
(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))))))))