HairBSDF, sample_f, cosTheta

Percentage Accurate: 99.5% → 99.5%

Time: 4.2s

Alternatives: 9

Speedup: 1.2×

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, 1.2× 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 99.5%

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

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

   3. lift-log.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

3. Applied rewrites 99.5%

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

Alternative 2: 96.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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 + exp(((-2.0e0) / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 96.2

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

4. Applied rewrites 96.2%

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

Alternative 3: 96.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	return 1.0f + logf(powf((u + expf((-2.0f / v))), 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 + log(((u + exp(((-2.0e0) / v))) ** v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in u around 0

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 96.2

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

4. Applied rewrites 96.2%

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

   1. lift-*.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. log-pow-rev N/A

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

   4. lower-log.f32 N/A

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

   5. lower-pow.f32 96.1

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

6. Applied rewrites 96.1%

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

Alternative 4: 91.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if v < 0.200000003

   1. Initial program 100.0%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

   2. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-expm1.f32 N/A

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

      7. lower-neg.f32 N/A

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

      8. lift-/.f32 69.5

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

   4. Applied rewrites 69.5%

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

Alternative 5: 91.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float u, float v) {
	float tmp;
	if (v <= 0.20000000298023224f) {
		tmp = 1.0f;
	} else {
		tmp = ((v * expm1f((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(v * expm1(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 100.0%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 93.1%

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

   if 0.200000003 < v

   1. Initial program 93.5%

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

   2. Taylor expanded in u around 0

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

      1. lower--.f32 N/A

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

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u \cdot \left(v \cdot e^{\mathsf{log1p}\left(-e^{\frac{-2}{v}}\right) \cdot 2 - \frac{-2}{v} \cdot 2}\right), -0.5, \mathsf{expm1}\left(-\frac{-2}{v}\right) \cdot v\right) \cdot u - 1} \]

   5. Taylor expanded in u around 0

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

      1. lower-*.f32 N/A

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

      2. lower-expm1.f32 N/A

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

      3. lift-/.f32 69.5

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

   7. Applied rewrites 69.5%

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

Alternative 6: 90.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))) < -0.400000006

   1. Initial program 93.7%

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

   2. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(1 - u\right) \cdot -2 + 1 \]

      3. lower-fma.f32 N/A

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

      4. lift--.f32 58.0

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

   4. Applied rewrites 58.0%

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

   if -0.400000006 < (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 92.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))) < -0.400000006

   1. Initial program 93.7%

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

   2. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(1 - u\right) \cdot -2 + 1 \]

      3. lower-fma.f32 N/A

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

      4. lift--.f32 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in u around 0

      \[\leadsto 2 \cdot u - \color{blue}{1} \]
   6. Step-by-step derivation

      1. lower--.f32 N/A

         \[\leadsto 2 \cdot u - 1 \]

      2. lower-*.f32 58.0

         \[\leadsto 2 \cdot u - 1 \]

   7. Applied rewrites 58.0%

      \[\leadsto 2 \cdot u - \color{blue}{1} \]

   if -0.400000006 < (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 92.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v))))))) < -0.400000006

   1. Initial program 93.7%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 49.5%

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

   if -0.400000006 < (+.f32 #s(literal 1 binary32) (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 #s(literal 1 binary32) u) (exp.f32 (/.f32 #s(literal -2 binary32) v)))))))

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 92.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 6.0% accurate, 15.0× 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 99.5%

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

2. Taylor expanded in u around 0

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

4. Applied rewrites 6.0%

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

Reproduce

herbie shell --seed 2025101 
(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))))))))