HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 12.5s

Alternatives: 14

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 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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. 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 99.5%

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

   \[\leadsto \mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right) \]
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 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto 1 + v \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right) \]
8. 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 99.5%

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

Alternative 5: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right) \end{array} \]

FPCore

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

C

float code(float u, float v) {
	return 1.0f + (v * logf((expm1f((-2.0f / v)) * -u)));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(expm1(Float32(Float32(-2.0) / v)) * Float32(-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. 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 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in u around inf 95.2%

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

   1. +-commutative 95.2%

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

   2. distribute-rgt-in 95.2%

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

   3. mul-1-neg 95.2%

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

   4. distribute-lft-neg-in 95.2%

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

   5. distribute-rgt-neg-out 95.2%

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

   6. *-lft-identity 95.2%

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

   7. remove-double-neg 95.2%

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

   8. neg-mul-1 95.2%

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

   9. distribute-rgt-in 95.2%

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

   10. metadata-eval 95.2%

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

   11. sub-neg 95.2%

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

   12. *-commutative 95.2%

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

   13. expm1-define 95.2%

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

9. Simplified 95.2%

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

10. Final simplification 95.2%

    \[\leadsto 1 + v \cdot \log \left(\mathsf{expm1}\left(\frac{-2}{v}\right) \cdot \left(-u\right)\right) \]
11. Add Preprocessing

Alternative 6: 91.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.5)
   1.0
   (+
    -1.0
    (*
     u
     (+
      (/
       (-
        (/
         (+
          (*
           -0.5
           (/
            (+
             (* u 9.333333333333334)
             (- (* 4.0 (- (* u 16.0) (* u 8.0))) (* u 32.0)))
            v))
          (* 0.5 (- (* u 8.0) (* u 16.0))))
         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 <= 0.5f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * ((((((-0.5f * (((u * 9.333333333333334f) + ((4.0f * ((u * 16.0f) - (u * 8.0f))) - (u * 32.0f))) / v)) + (0.5f * ((u * 8.0f) - (u * 16.0f)))) / 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 <= 0.5e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (((((((-0.5e0) * (((u * 9.333333333333334e0) + ((4.0e0 * ((u * 16.0e0) - (u * 8.0e0))) - (u * 32.0e0))) / v)) + (0.5e0 * ((u * 8.0e0) - (u * 16.0e0)))) / 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(0.5))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(u * Float32(9.333333333333334)) + Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0)))) - Float32(u * Float32(32.0)))) / v)) + Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0))))) / 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(0.5))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * ((((((single(-0.5) * (((u * single(9.333333333333334)) + ((single(4.0) * ((u * single(16.0)) - (u * single(8.0)))) - (u * single(32.0)))) / v)) + (single(0.5) * ((u * single(8.0)) - (u * single(16.0))))) / 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 < 0.5

   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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 92.1%

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

   if 0.5 < v

   1. Initial program 91.6%

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

   3. Taylor expanded in u around 0 77.1%

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

   4. Taylor expanded in v around -inf 79.6%

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

4. Final simplification 91.5%

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

Alternative 7: 91.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		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(Float32(u * Float32(2.0)) + Float32(Float32(u * Float32(Float32(4.0) + Float32(Float32(4.666666666666667) / v))) / v)) / v))));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 99.9%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   2. 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 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 92.1%

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

   if 0.5 < v

   1. Initial program 91.6%

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

   3. Taylor expanded in u around 0 77.1%

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

   4. Taylor expanded in v around -inf 79.6%

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

   5. Taylor expanded in u around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. mul-1-neg 79.6%

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

      3. associate-*r/ 79.6%

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

      4. metadata-eval 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 91.5%

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

Alternative 8: 91.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/
       (+
        2.0
        (-
         (/
          (+
           1.3333333333333333
           (+
            (* 0.5 (- (* u 8.0) (* u 16.0)))
            (/
             (+
              0.6666666666666666
              (*
               0.5
               (-
                (- (* u 32.0) (* 4.0 (- (* u 16.0) (* u 8.0))))
                (* u 9.333333333333334))))
             v)))
          v)
         (* u 2.0)))
       v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f + ((0.5f * ((u * 8.0f) - (u * 16.0f))) + ((0.6666666666666666f + (0.5f * (((u * 32.0f) - (4.0f * ((u * 16.0f) - (u * 8.0f)))) - (u * 9.333333333333334f)))) / v))) / v) - (u * 2.0f))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + ((0.5e0 * ((u * 8.0e0) - (u * 16.0e0))) + ((0.6666666666666666e0 + (0.5e0 * (((u * 32.0e0) - (4.0e0 * ((u * 16.0e0) - (u * 8.0e0)))) - (u * 9.333333333333334e0)))) / v))) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(Float32(Float32(1.3333333333333333) + Float32(Float32(Float32(0.5) * Float32(Float32(u * Float32(8.0)) - Float32(u * Float32(16.0)))) + Float32(Float32(Float32(0.6666666666666666) + Float32(Float32(0.5) * Float32(Float32(Float32(u * Float32(32.0)) - Float32(Float32(4.0) * Float32(Float32(u * Float32(16.0)) - Float32(u * Float32(8.0))))) - Float32(u * Float32(9.333333333333334))))) / v))) / v) - Float32(u * Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + ((single(0.5) * ((u * single(8.0)) - (u * single(16.0)))) + ((single(0.6666666666666666) + (single(0.5) * (((u * single(32.0)) - (single(4.0) * ((u * single(16.0)) - (u * single(8.0))))) - (u * single(9.333333333333334))))) / v))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.2%

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

   if 0.150000006 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 65.0%

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

   4. Taylor expanded in v around -inf 64.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + \left(0.5 \cdot \left(u \cdot 8 - u \cdot 16\right) + \frac{0.6666666666666666 + 0.5 \cdot \left(\left(u \cdot 32 - 4 \cdot \left(u \cdot 16 - u \cdot 8\right)\right) - u \cdot 9.333333333333334\right)}{v}\right)}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + u \cdot -4}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (*
     u
     (+
      2.0
      (/ (+ 2.0 (- (/ (+ 1.3333333333333333 (* u -4.0)) v) (* u 2.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + (((1.3333333333333333f + (u * -4.0f)) / 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 <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + (((1.3333333333333333e0 + (u * (-4.0e0))) / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		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(u * Float32(-4.0))) / 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(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + (((single(1.3333333333333333) + (u * single(-4.0))) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.2%

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

   if 0.150000006 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 65.0%

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

   4. Taylor expanded in v around -inf 62.0%

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

   5. Taylor expanded in u around 0 62.0%

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

      1. *-commutative 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333 + u \cdot -4}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333 + \frac{-0.6666666666666666}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    1.0
    (-
     (*
      u
      (-
       2.0
       (/ (+ -2.0 (/ (+ -1.3333333333333333 (/ -0.6666666666666666 v)) v)) v)))
     2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f - ((-2.0f + ((-1.3333333333333333f + (-0.6666666666666666f / v)) / v)) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 - (((-2.0e0) + (((-1.3333333333333333e0) + ((-0.6666666666666666e0) / v)) / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) - Float32(Float32(Float32(-2.0) + Float32(Float32(Float32(-1.3333333333333333) + Float32(Float32(-0.6666666666666666) / v)) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) - ((single(-2.0) + ((single(-1.3333333333333333) + (single(-0.6666666666666666) / v)) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.2%

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

   if 0.150000006 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 55.2%

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

   4. Taylor expanded in v around -inf 57.7%

      \[\leadsto 1 + \left(u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 2\right) \]
   5. Step-by-step derivation

      1. distribute-lft-in 57.7%

         \[\leadsto 1 + \left(\color{blue}{\left(u \cdot 2 + u \cdot \left(-1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)\right)} - 2\right) \]

      2. mul-1-neg 57.7%

         \[\leadsto 1 + \left(\left(u \cdot 2 + u \cdot \color{blue}{\left(-\frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)}\right) - 2\right) \]

      3. fma-neg 57.7%

         \[\leadsto 1 + \left(\left(u \cdot 2 + u \cdot \left(-\frac{\color{blue}{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v}, -2\right)}}{v}\right)\right) - 2\right) \]

      4. un-div-inv 57.7%

         \[\leadsto 1 + \left(\left(u \cdot 2 + u \cdot \left(-\frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + \color{blue}{\frac{0.6666666666666666}{v}}}{v}, -2\right)}{v}\right)\right) - 2\right) \]

      5. metadata-eval 57.7%

         \[\leadsto 1 + \left(\left(u \cdot 2 + u \cdot \left(-\frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}, \color{blue}{-2}\right)}{v}\right)\right) - 2\right) \]

   6. Applied egg-rr 57.7%

      \[\leadsto 1 + \left(\color{blue}{\left(u \cdot 2 + u \cdot \left(-\frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}, -2\right)}{v}\right)\right)} - 2\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 57.7%

         \[\leadsto 1 + \left(\color{blue}{u \cdot \left(2 + \left(-\frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}, -2\right)}{v}\right)\right)} - 2\right) \]

      2. unsub-neg 57.7%

         \[\leadsto 1 + \left(u \cdot \color{blue}{\left(2 - \frac{\mathsf{fma}\left(-1, \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}, -2\right)}{v}\right)} - 2\right) \]

      3. fma-undefine 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\color{blue}{-1 \cdot \frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v} + -2}}{v}\right) - 2\right) \]

      4. neg-mul-1 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\color{blue}{\left(-\frac{1.3333333333333333 + \frac{0.6666666666666666}{v}}{v}\right)} + -2}{v}\right) - 2\right) \]

      5. distribute-neg-frac 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\color{blue}{\frac{-\left(1.3333333333333333 + \frac{0.6666666666666666}{v}\right)}{v}} + -2}{v}\right) - 2\right) \]

      6. distribute-neg-in 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{\color{blue}{\left(-1.3333333333333333\right) + \left(-\frac{0.6666666666666666}{v}\right)}}{v} + -2}{v}\right) - 2\right) \]

      7. metadata-eval 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{\color{blue}{-1.3333333333333333} + \left(-\frac{0.6666666666666666}{v}\right)}{v} + -2}{v}\right) - 2\right) \]

      8. distribute-neg-frac 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \color{blue}{\frac{-0.6666666666666666}{v}}}{v} + -2}{v}\right) - 2\right) \]

      9. metadata-eval 57.7%

         \[\leadsto 1 + \left(u \cdot \left(2 - \frac{\frac{-1.3333333333333333 + \frac{\color{blue}{-0.6666666666666666}}{v}}{v} + -2}{v}\right) - 2\right) \]

   8. Simplified 57.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 - \frac{-2 + \frac{-1.3333333333333333 + \frac{-0.6666666666666666}{v}}{v}}{v}\right) - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+
    -1.0
    (* u (+ 2.0 (/ (+ 2.0 (- (/ 1.3333333333333333 v) (* u 2.0))) v))))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = -1.0f + (u * (2.0f + ((2.0f + ((1.3333333333333333f / v) - (u * 2.0f))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = (-1.0e0) + (u * (2.0e0 + ((2.0e0 + ((1.3333333333333333e0 / v) - (u * 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(-1.0) + Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(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(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(-1.0) + (u * (single(2.0) + ((single(2.0) + ((single(1.3333333333333333) / v) - (u * single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.2%

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

   if 0.150000006 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 65.0%

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

   4. Taylor expanded in v around -inf 62.0%

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

   5. Taylor expanded in u around 0 58.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot \left(2 + \frac{2 + \left(\frac{1.3333333333333333}{v} - u \cdot 2\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.3% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(u \cdot \left(2 + \frac{2 + \frac{1.3333333333333333}{v}}{v}\right) - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.15000000596046448)
   1.0
   (+ 1.0 (- (* u (+ 2.0 (/ (+ 2.0 (/ 1.3333333333333333 v)) v))) 2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.15000000596046448f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + ((u * (2.0f + ((2.0f + (1.3333333333333333f / v)) / v))) - 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(u, v)
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.15000000596046448e0) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 + ((u * (2.0e0 + ((2.0e0 + (1.3333333333333333e0 / v)) / v))) - 2.0e0)
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.15000000596046448))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(2.0) + Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / v)) / v))) - Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.15000000596046448))
		tmp = single(1.0);
	else
		tmp = single(1.0) + ((u * (single(2.0) + ((single(2.0) + (single(1.3333333333333333) / v)) / v))) - single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.150000006

   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 93.2%

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

   if 0.150000006 < v

   1. Initial program 92.4%

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

   3. Taylor expanded in u around 0 55.2%

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

   4. Taylor expanded in v around -inf 57.7%

      \[\leadsto 1 + \left(u \cdot \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \frac{1.3333333333333333 + 0.6666666666666666 \cdot \frac{1}{v}}{v} - 2}{v}\right)} - 2\right) \]

   5. Taylor expanded in v around inf 56.7%

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

      1. associate-*r/ 56.7%

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

      2. metadata-eval 56.7%

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

   7. Simplified 56.7%

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

Alternative 13: 87.1% accurate, 213.0× speedup

Math

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

FPCore

(FPCore (u v) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
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 87.7%

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

Alternative 14: 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 99.5%

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

3. Taylor expanded in u around 0 5.3%

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

Reproduce

herbie shell --seed 2024119 
(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))))))))