HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 8.6s

Alternatives: 9

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 \cdot v, \log \left({\left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)}^{2}\right), 1\right) \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (fma (* 0.5 v) (log (pow (fma (exp (/ -2.0 v)) (- 1.0 u) u) 2.0)) 1.0))

C

float code(float u, float v) {
	return fmaf((0.5f * v), logf(powf(fmaf(expf((-2.0f / v)), (1.0f - u), u), 2.0f)), 1.0f);
}

Julia

function code(u, v)
	return fma(Float32(Float32(0.5) * v), log((fma(exp(Float32(Float32(-2.0) / v)), Float32(Float32(1.0) - u), u) ^ Float32(2.0))), 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. *-lft-identity N/A

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

   4. *-lft-identity N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. log-pow-rev N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. log-pow N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

   1. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{0.5 \cdot v}, \log \left({\left(\mathsf{fma}\left(e^{\frac{-2}{v}}, 1 - u, u\right)\right)}^{2}\right), 1\right) \]
8. 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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, v)
	return fma(log(fma(Float32(1.0), 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

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. *-lft-identity N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. log-pow-rev N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. log-pow N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

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. Taylor expanded in u around 0

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

9. Applied rewrites 97.0%

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

Alternative 4: 91.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(2 + \frac{1.3333333333333333}{v \cdot v}\right) + \mathsf{fma}\left(\frac{u}{v \cdot v} \cdot 2.6666666666666665 - \left(\frac{4}{v \cdot v} + \frac{2}{v}\right), u, \frac{4}{v}\right)\right) - \frac{2}{v}\right) \cdot u - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (*
     (-
      (+
       (+ 2.0 (/ 1.3333333333333333 (* v v)))
       (fma
        (- (* (/ u (* v v)) 2.6666666666666665) (+ (/ 4.0 (* v v)) (/ 2.0 v)))
        u
        (/ 4.0 v)))
      (/ 2.0 v))
     u)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = ((((2.0f + (1.3333333333333333f / (v * v))) + fmaf((((u / (v * v)) * 2.6666666666666665f) - ((4.0f / (v * v)) + (2.0f / v))), u, (4.0f / v))) - (2.0f / v)) * u) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(1.3333333333333333) / Float32(v * v))) + fma(Float32(Float32(Float32(u / Float32(v * v)) * Float32(2.6666666666666665)) - Float32(Float32(Float32(4.0) / Float32(v * v)) + Float32(Float32(2.0) / v))), u, Float32(Float32(4.0) / v))) - Float32(Float32(2.0) / v)) * u) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.0%

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

   if 0.200000003 < v

   1. Initial program 93.1%

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

   3. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in v around -inf

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. associate-+r- N/A

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

   11. Applied rewrites 71.7%

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

   12. Taylor expanded in u around 0

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

   14. Applied rewrites 72.2%

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

Alternative 5: 91.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-u, \frac{4}{v \cdot v} + \frac{2}{v}, \left(2 + \frac{4}{v}\right) + \frac{1.3333333333333333}{v \cdot v}\right) - \frac{2}{v}\right) \cdot u - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.20000000298023224)
   1.0
   (-
    (*
     (-
      (fma
       (- u)
       (+ (/ 4.0 (* v v)) (/ 2.0 v))
       (+ (+ 2.0 (/ 4.0 v)) (/ 1.3333333333333333 (* v v))))
      (/ 2.0 v))
     u)
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = ((fmaf(-u, ((4.0f / (v * v)) + (2.0f / v)), ((2.0f + (4.0f / v)) + (1.3333333333333333f / (v * v)))) - (2.0f / v)) * u) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.20000000298023224))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(fma(Float32(-u), Float32(Float32(Float32(4.0) / Float32(v * v)) + Float32(Float32(2.0) / v)), Float32(Float32(Float32(2.0) + Float32(Float32(4.0) / v)) + Float32(Float32(1.3333333333333333) / Float32(v * v)))) - Float32(Float32(2.0) / v)) * u) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 92.0%

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

   if 0.200000003 < v

   1. Initial program 93.1%

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

   3. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in v around -inf

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. associate-+r- N/A

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

   11. Applied rewrites 71.7%

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

   12. Taylor expanded in u around 0

       \[\leadsto u \cdot \left(\left(2 + \left(-1 \cdot \left(u \cdot \left(2 \cdot \frac{1}{v} + 4 \cdot \frac{1}{{v}^{2}}\right)\right) + \left(\frac{4}{3} \cdot \frac{1}{{v}^{2}} + 4 \cdot \frac{1}{v}\right)\right)\right) - 2 \cdot \frac{1}{v}\right) - \color{blue}{1} \]
   13. Step-by-step derivation

   14. Applied rewrites 70.4%

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

Alternative 6: 91.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, u, \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{-v}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645)
   1.0
   (-
    (fma
     2.0
     u
     (/
      (-
       (* -2.0 u)
       (/ (fma 0.6666666666666666 (/ u v) (* 1.3333333333333333 u)) v))
      (- v)))
    1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = fmaf(2.0f, u, (((-2.0f * u) - (fmaf(0.6666666666666666f, (u / v), (1.3333333333333333f * u)) / v)) / -v)) - 1.0f;
	}
	return tmp;
}

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(fma(Float32(2.0), u, Float32(Float32(Float32(Float32(-2.0) * u) - Float32(fma(Float32(0.6666666666666666), Float32(u / v), Float32(Float32(1.3333333333333333) * u)) / v)) / Float32(-v))) - Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.7%

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

   if 0.400000006 < v

   1. Initial program 92.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in v around -inf

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

   8. Applied rewrites 67.4%

      \[\leadsto \mathsf{fma}\left(2, u, \frac{-2 \cdot u - \frac{\mathsf{fma}\left(0.6666666666666666, \frac{u}{v}, 1.3333333333333333 \cdot u\right)}{v}}{-v}\right) - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{u}{v} + u\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v)
 :precision binary32
 (if (<= v 0.4000000059604645) 1.0 (- (* 2.0 (+ (/ u v) u)) 1.0)))

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.4000000059604645f) {
		tmp = 1.0f;
	} else {
		tmp = (2.0f * ((u / v) + u)) - 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 <= 0.4000000059604645e0) then
        tmp = 1.0e0
    else
        tmp = (2.0e0 * ((u / v) + u)) - 1.0e0
    end if
    code = tmp
end function

Julia

function code(u, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.4000000059604645))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(u / v) + u)) - Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v)
	tmp = single(0.0);
	if (v <= single(0.4000000059604645))
		tmp = single(1.0);
	else
		tmp = (single(2.0) * ((u / v) + u)) - single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 91.7%

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

   if 0.400000006 < v

   1. Initial program 92.8%

      \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   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. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-expm1.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 67.0%

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

Alternative 8: 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 87.2%

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

Alternative 9: 5.9% 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 5.4%

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

Reproduce

herbie shell --seed 2024350 
(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))))))))