HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 6.8s

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.6

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 99.6

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

   6. lift-+.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.6

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 99.6

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in u around 0

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

    1. metadata-eval N/A

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

    2. distribute-frac-neg N/A

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

    3. metadata-eval N/A

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

    4. associate-*r/ N/A

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

    5. distribute-lft-neg-out N/A

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

    6. lower-exp.f32 N/A

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

    7. distribute-lft-neg-out N/A

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

    8. associate-*r/ N/A

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

    9. metadata-eval N/A

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

    10. distribute-frac-neg N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 96.5

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

11. Applied rewrites 96.5%

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

Alternative 4: 91.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{u}{v}}{v}, 2.6666666666666665, \frac{-2}{v}\right) - \frac{4}{v \cdot v}, u, \frac{4}{v}\right)\right) - \frac{2}{v}, u, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (fma
     (-
      (+
       (+ 2.0 (/ 1.3333333333333333 (* v v)))
       (fma
        (- (fma (/ (/ u v) v) 2.6666666666666665 (/ -2.0 v)) (/ 4.0 (* v v)))
        u
        (/ 4.0 v)))
      (/ 2.0 v))
     u
     -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf((((2.0f + (1.3333333333333333f / (v * v))) + fmaf((fmaf(((u / v) / v), 2.6666666666666665f, (-2.0f / v)) - (4.0f / (v * v))), u, (4.0f / v))) - (2.0f / v)), u, -2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))) + fma(Float32(fma(Float32(Float32(u / v) / v), Float32(2.6666666666666665), Float32(Float32(-2.0) / v)) - Float32(Float32(4.0) / Float32(v * v))), u, Float32(Float32(4.0) / v))) - Float32(Float32(2.0) / v)), u, Float32(-2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Applied rewrites 74.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-24, {\left(1 - u\right)}^{2}, \mathsf{fma}\left(8, 1 - u, 16 \cdot {\left(1 - u\right)}^{3}\right)\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(24 + -16 \cdot u, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(-16 \cdot u + 24, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. lower-fma.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   9. Taylor expanded in u around 0

      \[\leadsto 1 + \left(u \cdot \left(\left(2 + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + \left(4 \cdot \frac{1}{v} + u \cdot \left(\frac{8}{3} \cdot \frac{u}{{v}^{2}} - \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right) - 2 \cdot \frac{1}{v}\right) - \color{blue}{2}\right) \]

   11. Applied rewrites 75.0%

       \[\leadsto 1 + \mathsf{fma}\left(\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{u}{v}}{v}, 2.6666666666666665, \frac{-2}{v}\right) - \frac{4}{v \cdot v}, u, \frac{4}{v}\right)\right) - \frac{2}{v}, \color{blue}{u}, -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{u}{v}}{v}, 2.6666666666666665, \frac{-2}{v}\right) - \frac{4}{v \cdot v}, u, \frac{4}{v}\right)\right) - \frac{2}{v}, u, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot u\\ \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{t\_0 \cdot t\_0}{\left(u + 1\right) \cdot \left(u + 1\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u u))))
   (if (<= v 0.05000000074505806)
     1.0
     (+
      1.0
      (fma
       -2.0
       (- 1.0 u)
       (/
        (fma
         -0.16666666666666666
         (/ (* (fma (fma -16.0 u 24.0) u -8.0) u) v)
         (fma (/ (* t_0 t_0) (* (+ u 1.0) (+ u 1.0))) -2.0 (* 2.0 (- 1.0 u))))
        v))))))

C

float code(float u, float v) {
	float t_0 = 1.0f - (u * u);
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), (fmaf(-0.16666666666666666f, ((fmaf(fmaf(-16.0f, u, 24.0f), u, -8.0f) * u) / v), fmaf(((t_0 * t_0) / ((u + 1.0f) * (u + 1.0f))), -2.0f, (2.0f * (1.0f - u)))) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	t_0 = Float32(Float32(1.0) - Float32(u * u))
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(-0.16666666666666666), Float32(Float32(fma(fma(Float32(-16.0), u, Float32(24.0)), u, Float32(-8.0)) * u) / v), fma(Float32(Float32(t_0 * t_0) / Float32(Float32(u + Float32(1.0)) * Float32(u + Float32(1.0)))), Float32(-2.0), Float32(Float32(2.0) * Float32(Float32(1.0) - u)))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Applied rewrites 74.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-24, {\left(1 - u\right)}^{2}, \mathsf{fma}\left(8, 1 - u, 16 \cdot {\left(1 - u\right)}^{3}\right)\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(24 + -16 \cdot u, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(-16 \cdot u + 24, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. lower-fma.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. unpow2 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. lift--.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(1 - u\right) \cdot \left(1 - u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. flip-- N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{1 \cdot 1 - u \cdot u}{1 + u} \cdot \left(1 - u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. lift--.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{1 \cdot 1 - u \cdot u}{1 + u} \cdot \left(1 - u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. flip-- N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{1 \cdot 1 - u \cdot u}{1 + u} \cdot \frac{1 \cdot 1 - u \cdot u}{1 + u}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. frac-times N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 \cdot 1 - u \cdot u\right) \cdot \left(1 \cdot 1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-/.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 \cdot 1 - u \cdot u\right) \cdot \left(1 \cdot 1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 \cdot 1 - u \cdot u\right) \cdot \left(1 \cdot 1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 \cdot 1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      11. lower--.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 \cdot 1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 \cdot 1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      13. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      14. lower--.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      15. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      16. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(1 + u\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      17. +-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(u + 1\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      18. lower-+.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(u + 1\right) \cdot \left(1 + u\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      19. +-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(u + 1\right) \cdot \left(u + 1\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      20. lower-+.f32 74.9

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(u + 1\right) \cdot \left(u + 1\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   10. Applied rewrites 74.9%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(u + 1\right) \cdot \left(u + 1\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\frac{\left(1 - u \cdot u\right) \cdot \left(1 - u \cdot u\right)}{\left(u + 1\right) \cdot \left(u + 1\right)}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot \left(u \cdot u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       -0.16666666666666666
       (/ (* (fma (fma -16.0 u 24.0) u -8.0) u) v)
       (fma
        (* (- (+ (/ 1.0 (* u u)) 1.0) (/ 2.0 u)) (* u u))
        -2.0
        (* 2.0 (- 1.0 u))))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), (fmaf(-0.16666666666666666f, ((fmaf(fmaf(-16.0f, u, 24.0f), u, -8.0f) * u) / v), fmaf(((((1.0f / (u * u)) + 1.0f) - (2.0f / u)) * (u * u)), -2.0f, (2.0f * (1.0f - u)))) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(-0.16666666666666666), Float32(Float32(fma(fma(Float32(-16.0), u, Float32(24.0)), u, Float32(-8.0)) * u) / v), fma(Float32(Float32(Float32(Float32(Float32(1.0) / Float32(u * u)) + Float32(1.0)) - Float32(Float32(2.0) / u)) * Float32(u * u)), Float32(-2.0), Float32(Float32(2.0) * Float32(Float32(1.0) - u)))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Applied rewrites 74.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-24, {\left(1 - u\right)}^{2}, \mathsf{fma}\left(8, 1 - u, 16 \cdot {\left(1 - u\right)}^{3}\right)\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(24 + -16 \cdot u, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(-16 \cdot u + 24, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. lower-fma.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   9. Taylor expanded in u around inf

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({u}^{2} \cdot \left(\left(1 + \frac{1}{{u}^{2}}\right) - 2 \cdot \frac{1}{u}\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(1 + \frac{1}{{u}^{2}}\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       2. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(1 + \frac{1}{{u}^{2}}\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       3. lower--.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(1 + \frac{1}{{u}^{2}}\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       4. +-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{{u}^{2}} + 1\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       5. lower-+.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{{u}^{2}} + 1\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       6. lower-/.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{{u}^{2}} + 1\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       7. unpow2 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       8. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - 2 \cdot \frac{1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       9. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2 \cdot 1}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       10. metadata-eval N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       11. lower-/.f32 N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot {u}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       12. unpow2 N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot \left(u \cdot u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

       13. lower-*.f32 74.8

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot \left(u \cdot u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   11. Applied rewrites 74.8%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot \left(u \cdot u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(\left(\left(\frac{1}{u \cdot u} + 1\right) - \frac{2}{u}\right) \cdot \left(u \cdot u\right), -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(\frac{2}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       -0.16666666666666666
       (/ (* (fma (fma -16.0 u 24.0) u -8.0) u) v)
       (* (* (- (/ 2.0 u) 2.0) u) u))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), (fmaf(-0.16666666666666666f, ((fmaf(fmaf(-16.0f, u, 24.0f), u, -8.0f) * u) / v), ((((2.0f / u) - 2.0f) * u) * u)) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(-0.16666666666666666), Float32(Float32(fma(fma(Float32(-16.0), u, Float32(24.0)), u, Float32(-8.0)) * u) / v), Float32(Float32(Float32(Float32(Float32(2.0) / u) - Float32(2.0)) * u) * u)) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Applied rewrites 74.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-24, {\left(1 - u\right)}^{2}, \mathsf{fma}\left(8, 1 - u, 16 \cdot {\left(1 - u\right)}^{3}\right)\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(24 + -16 \cdot u, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(-16 \cdot u + 24, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. lower-fma.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   9. Taylor expanded in u around inf

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, {u}^{2} \cdot \left(2 \cdot \frac{1}{u} - 2\right)\right)}{v}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot \frac{1}{u} - 2\right) \cdot {u}^{2}\right)}{v}\right) \]

       2. unpow2 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot \frac{1}{u} - 2\right) \cdot \left(u \cdot u\right)\right)}{v}\right) \]

       3. associate-*r* N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(2 \cdot \frac{1}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       4. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(2 \cdot \frac{1}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(2 \cdot \frac{1}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       6. lower--.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(2 \cdot \frac{1}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       7. associate-*r/ N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(\frac{2 \cdot 1}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       8. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(\frac{2}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       9. lower-/.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(\frac{2}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]

   11. Applied rewrites 74.8%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(\frac{2}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(\left(\frac{2}{u} - 2\right) \cdot u\right) \cdot u\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(-2, u, 2\right) \cdot u\right)}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (fma
       -0.16666666666666666
       (/ (* (fma (fma -16.0 u 24.0) u -8.0) u) v)
       (* (fma -2.0 u 2.0) u))
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), (fmaf(-0.16666666666666666f, ((fmaf(fmaf(-16.0f, u, 24.0f), u, -8.0f) * u) / v), (fmaf(-2.0f, u, 2.0f) * u)) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(fma(Float32(-0.16666666666666666), Float32(Float32(fma(fma(Float32(-16.0), u, Float32(24.0)), u, Float32(-8.0)) * u) / v), Float32(fma(Float32(-2.0), u, Float32(2.0)) * u)) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Applied rewrites 74.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-24, {\left(1 - u\right)}^{2}, \mathsf{fma}\left(8, 1 - u, 16 \cdot {\left(1 - u\right)}^{3}\right)\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(24 + -16 \cdot u, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(-16 \cdot u + 24, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. lower-fma.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   9. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, u \cdot \left(2 + -2 \cdot u\right)\right)}{v}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 + -2 \cdot u\right) \cdot u\right)}{v}\right) \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 - \left(\mathsf{neg}\left(-2\right)\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       3. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot 1 - \left(\mathsf{neg}\left(-2\right)\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       4. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot 1 - 2 \cdot u\right) \cdot u\right)}{v}\right) \]

       5. distribute-lft-out-- N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot \left(1 - u\right)\right) \cdot u\right)}{v}\right) \]

       6. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot \left(1 - u\right)\right) \cdot u\right)}{v}\right) \]

       7. distribute-lft-out-- N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 \cdot 1 - 2 \cdot u\right) \cdot u\right)}{v}\right) \]

       8. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 - 2 \cdot u\right) \cdot u\right)}{v}\right) \]

       9. metadata-eval N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 - \left(\mathsf{neg}\left(-2\right)\right) \cdot u\right) \cdot u\right)}{v}\right) \]

       10. fp-cancel-sign-sub-inv N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(2 + -2 \cdot u\right) \cdot u\right)}{v}\right) \]

       11. +-commutative N/A

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \left(-2 \cdot u + 2\right) \cdot u\right)}{v}\right) \]

       12. lower-fma.f32 74.8

           \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(-2, u, 2\right) \cdot u\right)}{v}\right) \]

   11. Applied rewrites 74.8%

       \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(-2, u, 2\right) \cdot u\right)}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left(-2, u, 2\right) \cdot u\right)}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\left(\frac{1.3333333333333333}{v} - \left(\frac{4}{v} - -2\right) \cdot u\right) - -2\right) \cdot u}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+
    1.0
    (fma
     -2.0
     (- 1.0 u)
     (/
      (* (- (- (/ 1.3333333333333333 v) (* (- (/ 4.0 v) -2.0) u)) -2.0) u)
      v)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(-2.0f, (1.0f - u), (((((1.3333333333333333f / v) - (((4.0f / v) - -2.0f) * u)) - -2.0f) * u) / v));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(-2.0), Float32(Float32(1.0) - u), Float32(Float32(Float32(Float32(Float32(Float32(1.3333333333333333) / v) - Float32(Float32(Float32(Float32(4.0) / v) - Float32(-2.0)) * u)) - Float32(-2.0)) * u) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in v around -inf

      \[\leadsto 1 + \color{blue}{\left(-2 \cdot \left(1 - u\right) + -1 \cdot \frac{\frac{-1}{2} \cdot \left(-4 \cdot {\left(1 - u\right)}^{2} + 4 \cdot \left(1 - u\right)\right) + \frac{1}{6} \cdot \frac{-24 \cdot {\left(1 - u\right)}^{2} + \left(8 \cdot \left(1 - u\right) + 16 \cdot {\left(1 - u\right)}^{3}\right)}{v}}{v}\right)} \]

   5. Applied rewrites 74.8%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(-24, {\left(1 - u\right)}^{2}, \mathsf{fma}\left(8, 1 - u, 16 \cdot {\left(1 - u\right)}^{3}\right)\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right)} \]

   6. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{u \cdot \left(u \cdot \left(24 + -16 \cdot u\right) - 8\right)}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) - 8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(u \cdot \left(24 + -16 \cdot u\right) + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      5. *-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + \left(\mathsf{neg}\left(8\right)\right) \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      6. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8 \cdot 1\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      7. metadata-eval N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\left(\left(24 + -16 \cdot u\right) \cdot u + -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(24 + -16 \cdot u, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      9. +-commutative N/A

         \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\mathsf{fma}\left(-16 \cdot u + 24, u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

      10. lower-fma.f32 74.8

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   8. Applied rewrites 74.8%

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-16, u, 24\right), u, -8\right) \cdot u}{v}, \mathsf{fma}\left({\left(1 - u\right)}^{2}, -2, 2 \cdot \left(1 - u\right)\right)\right)}{v}\right) \]

   9. Taylor expanded in u around 0

      \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{u \cdot \left(2 + \left(-1 \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right)\right) + \frac{4}{3} \cdot \frac{1}{v}\right)\right)}{v}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(2 + \left(-1 \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right)\right) + \frac{4}{3} \cdot \frac{1}{v}\right)\right) \cdot u}{v}\right) \]

       2. lower-*.f32 N/A

          \[\leadsto 1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(2 + \left(-1 \cdot \left(u \cdot \left(2 + 4 \cdot \frac{1}{v}\right)\right) + \frac{4}{3} \cdot \frac{1}{v}\right)\right) \cdot u}{v}\right) \]

   11. Applied rewrites 66.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(-2, 1 - u, \frac{\left(\left(\frac{1.3333333333333333}{v} - \left(\frac{4}{v} - -2\right) \cdot u\right) - -2\right) \cdot u}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\mathsf{fma}\left(-2, u, 2\right) \cdot u}{v} - \mathsf{fma}\left(-2, u, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+ 1.0 (- (/ (* (fma -2.0 u 2.0) u) v) (fma -2.0 u 2.0)))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + (((fmaf(-2.0f, u, 2.0f) * u) / v) - fmaf(-2.0f, u, 2.0f));
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(Float32(fma(Float32(-2.0), u, Float32(2.0)) * u) / v) - fma(Float32(-2.0), u, Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 93.8

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. lower--.f32 N/A

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 64.4

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

   10. Applied rewrites 64.4%

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

Alternative 11: 90.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, u, 2\right)}{v} - -2, u, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806)
   1.0
   (+ 1.0 (fma (- (/ (fma -2.0 u 2.0) v) -2.0) u -2.0))))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f + fmaf(((fmaf(-2.0f, u, 2.0f) / v) - -2.0f), u, -2.0f);
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) + fma(Float32(Float32(fma(Float32(-2.0), u, Float32(2.0)) / v) - Float32(-2.0)), u, Float32(-2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 93.8

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. lower--.f32 N/A

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in u around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

   10. Applied rewrites 64.4%

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

Alternative 12: 90.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.05000000074505806:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{u}{v} + u\right) \cdot 2 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.05000000074505806) 1.0 (- (* (+ (/ u v) u) 2.0) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.05000000074505806f) {
		tmp = 1.0f;
	} else {
		tmp = (((u / v) + u) * 2.0f) - 1.0f;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u, v)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.05000000074505806e0) then
        tmp = 1.0e0
    else
        tmp = (((u / v) + u) * 2.0e0) - 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.05000000074505806))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(u / v) + u) * Float32(2.0)) - Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.05000000074505806))
		tmp = single(1.0);
	else
		tmp = (((u / v) + u) * single(2.0)) - single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.0500000007

   1. Initial program 100.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

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

   if 0.0500000007 < v

   1. Initial program 93.8%

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

   3. Taylor expanded in u around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. rec-exp N/A

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

      9. distribute-frac-neg N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. lower-expm1.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f32 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in v around inf

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

      1. lower--.f32 N/A

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

      2. distribute-lft-out N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f32 N/A

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

      7. lower-/.f32 60.1

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

   8. Applied rewrites 60.1%

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

Alternative 13: 87.2% accurate, 231.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 89.1%

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

Alternative 14: 5.8% accurate, 231.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 5.6%

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

Reproduce

herbie shell --seed 2025026 
(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))))))))