HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 13.2s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.7%

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

3. Simplified 99.7%

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

Alternative 2: 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 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.7%

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

3. Simplified 99.7%

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

   1. fma-undefine 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. *-commutative 99.7%

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

   3. fma-define 99.7%

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

5. Simplified 99.7%

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

Alternative 4: 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 99.7%

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

Alternative 5: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in u around 0 97.3%

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

Alternative 6: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in u around inf 95.9%

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

   1. +-commutative 95.9%

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

   2. distribute-rgt-in 95.9%

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

   3. mul-1-neg 95.9%

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

   4. distribute-lft-neg-in 95.9%

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

   5. distribute-rgt-neg-out 95.9%

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

   6. *-lft-identity 95.9%

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

   7. remove-double-neg 95.9%

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

   8. neg-mul-1 95.9%

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

   9. distribute-rgt-in 95.9%

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

   10. metadata-eval 95.9%

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

   11. sub-neg 95.9%

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

   12. distribute-lft-neg-in 95.9%

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

   13. distribute-rgt-neg-in 95.9%

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

   14. expm1-define 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in v around inf 88.1%

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

Alternative 7: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (/
       (-
        (/
         (-
          (* 0.5 (- (* u 8.0) (* u 16.0)))
          (* u (/ (+ 4.666666666666667 (/ 4.0 v)) v)))
         v)
        (* u 2.0))
       v)
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v))))))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((((((0.5f * ((u * 8.0f) - (u * 16.0f))) - (u * ((4.666666666666667f + (4.0f / v)) / v))) / v) - (u * 2.0f)) / v) + (v * (-1.0f + (1.0f / expf((-2.0f / v)))))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((((((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) - (u * ((4.666666666666667e0 + (4.0e0 / v)) / v))) / v) - (u * 2.0e0)) / v) + (v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v)))))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) - Float32(u * Float32(Float32(Float32(4.666666666666667) + Float32(Float32(4.0) / v)) / v))) / v) - Float32(u * Float32(2.0))) / v) + Float32(v * Float32(Float32(-1.0) + Float32(Float32(1.0) / exp(Float32(Float32(-2.0) / v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((((((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) - (u * ((single(4.666666666666667) + (single(4.0) / v)) / v))) / v) - (u * single(2.0))) / v) + (v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. 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. Taylor expanded in u around inf 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-rgt-in 99.6%

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

      3. mul-1-neg 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. distribute-rgt-neg-out 99.6%

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

      6. *-lft-identity 99.6%

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

      7. remove-double-neg 99.6%

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

      8. neg-mul-1 99.6%

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

      9. distribute-rgt-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. sub-neg 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. distribute-rgt-neg-in 99.6%

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

      14. expm1-define 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.7%

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

      3. +-commutative 94.7%

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

      4. fma-define 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in u around 0 77.2%

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

   6. Taylor expanded in v around -inf 74.9%

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

   7. Taylor expanded in u around 0 74.9%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{-1 \cdot \frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
   8. Step-by-step derivation

      1. mul-1-neg 74.9%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\left(-\frac{u \cdot \left(4.666666666666667 + 4 \cdot \frac{1}{v}\right)}{v}\right)} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      2. associate-/l* 74.9%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-\color{blue}{u \cdot \frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}}\right) + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      3. distribute-lft-neg-in 74.9%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\left(-u\right) \cdot \frac{4.666666666666667 + 4 \cdot \frac{1}{v}}{v}} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      4. associate-*r/ 74.9%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-u\right) \cdot \frac{4.666666666666667 + \color{blue}{\frac{4 \cdot 1}{v}}}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

      5. metadata-eval 74.9%

         \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-u\right) \cdot \frac{4.666666666666667 + \frac{\color{blue}{4}}{v}}{v} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]

   9. Simplified 74.9%

      \[\leadsto u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\color{blue}{\left(-u\right) \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}} + 0.5 \cdot \left(8 \cdot u - 16 \cdot u\right)}{v} + 2 \cdot u}{v} + v \cdot \left(\frac{1}{e^{\frac{-2}{v}}} - 1\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(\frac{\frac{0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) - u \cdot \frac{4.666666666666667 + \frac{4}{v}}{v}}{v} - u \cdot 2}{v} + v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - -0.5 \cdot \frac{u \cdot -4 + \frac{u}{v} \cdot -8}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.25)
   1.0
   (+
    -1.0
    (*
     u
     (-
      (* v (+ -1.0 (/ 1.0 (exp (/ -2.0 v)))))
      (* -0.5 (/ (+ (* u -4.0) (* (/ u v) -8.0)) v)))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.25f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((v * (-1.0f + (1.0f / expf((-2.0f / v))))) - (-0.5f * (((u * -4.0f) + ((u / v) * -8.0f)) / v))));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.25e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * ((v * ((-1.0e0) + (1.0e0 / exp(((-2.0e0) / v))))) - ((-0.5e0) * (((u * (-4.0e0)) + ((u / v) * (-8.0e0))) / v))))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.25))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(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 / v) * Float32(-8.0))) / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.25))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((v * (single(-1.0) + (single(1.0) / exp((single(-2.0) / v))))) - (single(-0.5) * (((u * single(-4.0)) + ((u / v) * single(-8.0))) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.25

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. 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. Taylor expanded in u around inf 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-rgt-in 99.6%

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

      3. mul-1-neg 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. distribute-rgt-neg-out 99.6%

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

      6. *-lft-identity 99.6%

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

      7. remove-double-neg 99.6%

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

      8. neg-mul-1 99.6%

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

      9. distribute-rgt-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. sub-neg 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. distribute-rgt-neg-in 99.6%

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

      14. expm1-define 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in v around 0 91.9%

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

   if 0.25 < v

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. fma-define 94.7%

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

      3. +-commutative 94.7%

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

      4. fma-define 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in u around 0 77.2%

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

   6. Taylor expanded in v around -inf 72.9%

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

      1. mul-1-neg 72.9%

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

      2. distribute-neg-frac2 72.9%

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

      3. associate--l+ 72.9%

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

      4. *-commutative 72.9%

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

      5. associate-*r/ 72.9%

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

      6. associate-*r/ 72.9%

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

      7. div-sub 72.9%

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

      8. distribute-rgt-out-- 72.9%

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

      9. metadata-eval 72.9%

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

      10. *-commutative 72.9%

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

      11. associate-*r/ 72.9%

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

      12. *-commutative 72.9%

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

   8. Simplified 72.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(v \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right) - -0.5 \cdot \frac{u \cdot -4 + \frac{u}{v} \cdot -8}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.100000001

   1. Initial program 100.0%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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. Taylor expanded in u around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. distribute-rgt-neg-out 100.0%

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

      6. *-lft-identity 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. sub-neg 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. expm1-define 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in v around 0 92.6%

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

   if 0.100000001 < v

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. fma-define 95.0%

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

      3. +-commutative 95.0%

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

      4. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in u around 0 68.7%

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

   6. Taylor expanded in v around -inf 61.7%

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

   7. Taylor expanded in u around 0 62.3%

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

4. Final simplification 91.0%

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

Alternative 10: 86.6% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in u around inf 95.9%

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

   1. +-commutative 95.9%

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

   2. distribute-rgt-in 95.9%

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

   3. mul-1-neg 95.9%

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

   4. distribute-lft-neg-in 95.9%

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

   5. distribute-rgt-neg-out 95.9%

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

   6. *-lft-identity 95.9%

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

   7. remove-double-neg 95.9%

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

   8. neg-mul-1 95.9%

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

   9. distribute-rgt-in 95.9%

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

   10. metadata-eval 95.9%

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

   11. sub-neg 95.9%

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

   12. distribute-lft-neg-in 95.9%

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

   13. distribute-rgt-neg-in 95.9%

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

   14. expm1-define 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in v around 0 87.8%

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

Alternative 11: 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 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. +-commutative 99.7%

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

   4. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in u around 0 5.0%

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

Reproduce

herbie shell --seed 2024185 
(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))))))))