HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.4%

Time: 12.7s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in u around inf 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float t_0 = expf((-2.0f / v));
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f + (v * logf((u * (1.0f + t_0))));
	} else {
		tmp = (u * ((v * (-1.0f - (-1.0f / t_0))) - (-0.5f * (((u * -4.0f) + ((u * -8.0f) / v)) / v)))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = exp(((-2.0e0) / v))
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0 + (v * log((u * (1.0e0 + t_0))))
    else
        tmp = (u * ((v * ((-1.0e0) - ((-1.0e0) / t_0))) - ((-0.5e0) * (((u * (-4.0e0)) + ((u * (-8.0e0)) / v)) / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	t_0 = exp(Float32(Float32(-2.0) / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(Float32(1.0) + t_0)))));
	else
		tmp = Float32(Float32(u * Float32(Float32(v * Float32(Float32(-1.0) - Float32(Float32(-1.0) / t_0))) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(u * Float32(-8.0)) / v)) / v)))) + Float32(-1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = exp((single(-2.0) / v));
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0) + (v * log((u * (single(1.0) + t_0))));
	else
		tmp = (u * ((v * (single(-1.0) - (single(-1.0) / t_0))) - (single(-0.5) * (((u * single(-4.0)) + ((u * single(-8.0)) / v)) / v)))) + single(-1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in u around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   5. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. fma-define 99.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 99.9%

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

      5. sqr-neg 99.9%

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

      6. sqrt-unprod 99.9%

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

      7. add-sqr-sqrt 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-rgt-out 99.9%

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

      4. +-commutative 99.9%

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

   9. Simplified 99.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in v around -inf 72.8%

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

      1. mul-1-neg 72.8%

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

      2. associate--l+ 72.8%

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

      3. *-commutative 72.8%

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

      4. associate-*r/ 72.8%

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

      5. associate-*r/ 72.8%

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

      6. div-sub 72.8%

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

      7. distribute-rgt-out-- 72.8%

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

      8. metadata-eval 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 97.8%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 + v \cdot \log \left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(-1 - \frac{-1}{e^{\frac{-2}{v}}}\right) - -0.5 \cdot \frac{u \cdot -4 + \frac{u \cdot -8}{v}}{v}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   (+ 1.0 (* v (log (* u (- (expm1 (/ -2.0 v)))))))
   (+
    (*
     u
     (-
      (* v (- -1.0 (/ -1.0 (exp (/ -2.0 v)))))
      (* -0.5 (/ (+ (* u -4.0) (/ (* u -8.0) v)) v))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f + (v * logf((u * -expm1f((-2.0f / v)))));
	} else {
		tmp = (u * ((v * (-1.0f - (-1.0f / expf((-2.0f / v))))) - (-0.5f * (((u * -4.0f) + ((u * -8.0f) / v)) / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) + Float32(v * log(Float32(u * Float32(-expm1(Float32(Float32(-2.0) / v)))))));
	else
		tmp = Float32(Float32(u * Float32(Float32(v * Float32(Float32(-1.0) - Float32(Float32(-1.0) / exp(Float32(Float32(-2.0) / v))))) - Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(-4.0)) + Float32(Float32(u * Float32(-8.0)) / v)) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   3. Taylor expanded in u around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto 1 + v \cdot \log \left(\color{blue}{\left(0 - u \cdot e^{\frac{-2}{v}}\right)} + u\right) \]

      4. associate-+l- 99.9%

         \[\leadsto 1 + v \cdot \log \color{blue}{\left(0 - \left(u \cdot e^{\frac{-2}{v}} - u\right)\right)} \]

      5. sub-neg 99.9%

         \[\leadsto 1 + v \cdot \log \left(0 - \color{blue}{\left(u \cdot e^{\frac{-2}{v}} + \left(-u\right)\right)}\right) \]

      6. mul-1-neg 99.9%

         \[\leadsto 1 + v \cdot \log \left(0 - \left(u \cdot e^{\frac{-2}{v}} + \color{blue}{-1 \cdot u}\right)\right) \]

      7. *-commutative 99.9%

         \[\leadsto 1 + v \cdot \log \left(0 - \left(u \cdot e^{\frac{-2}{v}} + \color{blue}{u \cdot -1}\right)\right) \]

      8. distribute-lft-in 99.9%

         \[\leadsto 1 + v \cdot \log \left(0 - \color{blue}{u \cdot \left(e^{\frac{-2}{v}} + -1\right)}\right) \]

      9. metadata-eval 99.9%

         \[\leadsto 1 + v \cdot \log \left(0 - u \cdot \left(e^{\frac{-2}{v}} + \color{blue}{\left(-1\right)}\right)\right) \]

      10. sub-neg 99.9%

          \[\leadsto 1 + v \cdot \log \left(0 - u \cdot \color{blue}{\left(e^{\frac{-2}{v}} - 1\right)}\right) \]

      11. expm1-define 99.9%

          \[\leadsto 1 + v \cdot \log \left(0 - u \cdot \color{blue}{\mathsf{expm1}\left(\frac{-2}{v}\right)}\right) \]

      12. neg-sub0 99.9%

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

      13. distribute-rgt-neg-in 99.9%

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

   8. Simplified 99.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in v around -inf 72.8%

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

      1. mul-1-neg 72.8%

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

      2. associate--l+ 72.8%

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

      3. *-commutative 72.8%

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

      4. associate-*r/ 72.8%

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

      5. associate-*r/ 72.8%

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

      6. div-sub 72.8%

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

      7. distribute-rgt-out-- 72.8%

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

      8. metadata-eval 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1 + v \cdot \log \left(u \cdot \left(-\mathsf{expm1}\left(\frac{-2}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \left(-1 - \frac{-1}{e^{\frac{-2}{v}}}\right) - -0.5 \cdot \frac{u \cdot -4 + \frac{u \cdot -8}{v}}{v}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 6: 91.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (+
    (*
     u
     (-
      (* v (- -1.0 (/ -1.0 (exp (/ -2.0 v)))))
      (* -0.5 (/ (+ (* u -4.0) (/ (* u -8.0) v)) v))))
    -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (u * ((v * (-1.0f - (-1.0f / expf((-2.0f / v))))) - (-0.5f * (((u * -4.0f) + ((u * -8.0f) / v)) / v)))) + -1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.20000000298023224e0) then
        tmp = 1.0e0
    else
        tmp = (u * ((v * ((-1.0e0) - ((-1.0e0) / exp(((-2.0e0) / v))))) - ((-0.5e0) * (((u * (-4.0e0)) + ((u * (-8.0e0)) / v)) / v)))) + (-1.0e0)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in v around -inf 72.8%

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

      1. mul-1-neg 72.8%

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

      2. associate--l+ 72.8%

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

      3. *-commutative 72.8%

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

      4. associate-*r/ 72.8%

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

      5. associate-*r/ 72.8%

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

      6. div-sub 72.8%

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

      7. distribute-rgt-out-- 72.8%

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

      8. metadata-eval 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 91.3%

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

Alternative 7: 91.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in v around inf 70.0%

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

4. Final simplification 91.1%

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

Alternative 8: 91.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 65.3%

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

4. Final simplification 90.7%

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

Alternative 9: 91.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224) 1.0 (+ (* v (* u (expm1 (/ 2.0 v)))) -1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = (v * (u * expm1f((2.0f / v)))) + -1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(v * Float32(u * expm1(Float32(Float32(2.0) / v)))) + Float32(-1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in u around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. rec-exp 65.3%

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

      3. expm1-undefine 65.3%

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

      4. associate-*l* 65.3%

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

      5. expm1-undefine 65.3%

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

      6. rec-exp 65.3%

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

      7. *-commutative 65.3%

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

      8. rec-exp 65.3%

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

      9. expm1-undefine 65.3%

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

      10. distribute-neg-frac 65.3%

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

      11. metadata-eval 65.3%

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

   6. Simplified 65.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(u \cdot \mathsf{expm1}\left(\frac{2}{v}\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(u, v)
	t_0 = Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(Float32(0.6666666666666666) + Float32(Float32(0.5) * Float32(Float32(Float32(Float32(4.0) * t_0) + Float32(u * Float32(32.0))) - Float32(u * Float32(9.333333333333334))))) / v) + Float32(Float32(0.5) * t_0))) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	t_0 = (u * single(8.0)) - (u * single(16.0));
	tmp = single(0.0);
	if (v <= single(0.20000000298023224))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + (((single(0.6666666666666666) + (single(0.5) * (((single(4.0) * t_0) + (u * single(32.0))) - (u * single(9.333333333333334))))) / v) + (single(0.5) * t_0))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in v around -inf 63.5%

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

4. Final simplification 90.6%

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

Alternative 11: 91.0% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in u around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. rec-exp 65.3%

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

      3. expm1-undefine 65.3%

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

      4. associate-*l* 65.3%

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

      5. expm1-undefine 65.3%

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

      6. rec-exp 65.3%

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

      7. *-commutative 65.3%

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

      8. rec-exp 65.3%

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

      9. expm1-undefine 65.3%

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

      10. distribute-neg-frac 65.3%

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

      11. metadata-eval 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in v around -inf 60.8%

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

4. Final simplification 90.4%

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

Alternative 12: 90.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in v around -inf 60.5%

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

4. Final simplification 90.4%

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

Alternative 13: 90.6% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in v around 0 92.9%

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

   if 0.200000003 < v

   1. Initial program 94.5%

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

   3. Taylor expanded in u around 0 73.7%

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

   4. Taylor expanded in u around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. rec-exp 65.3%

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

      3. expm1-undefine 65.3%

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

      4. associate-*l* 65.3%

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

      5. expm1-undefine 65.3%

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

      6. rec-exp 65.3%

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

      7. *-commutative 65.3%

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

      8. rec-exp 65.3%

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

      9. expm1-undefine 65.3%

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

      10. distribute-neg-frac 65.3%

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

      11. metadata-eval 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in v around inf 53.8%

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

      1. distribute-lft-out 53.8%

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

   9. Simplified 53.8%

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

4. Final simplification 89.8%

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

Alternative 14: 86.7% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in v around 0 86.6%

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

Alternative 15: 5.9% accurate, 213.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u around 0 5.7%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))