HairBSDF, Mp, lower

Percentage Accurate: 99.5% → 99.5%

Time: 9.6s

Alternatives: 4

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 57.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\\ t_1 := -1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)\\ \mathsf{fma}\left(e^{0.6931}, e^{\log \left(0.5 \cdot \frac{1}{v}\right) - \frac{-1 \cdot \mathsf{fma}\left(-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v, -1 \cdot v\right)}{t\_1}}, \mathsf{fma}\left(cosTheta\_O, \frac{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot t\_0}{t\_1} \cdot 0.5, \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot t\_0}{v}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (/
          (exp (- (+ (+ 0.6931 (log 0.5)) (* (log v) -1.0)) (/ 1.0 v)))
          (exp (/ (* sinTheta_O sinTheta_i) v))))
        (t_1 (* -1.0 (* (* -1.0 v) v))))
   (fma
    (exp 0.6931)
    (exp
     (-
      (log (* 0.5 (/ 1.0 v)))
      (/ (* -1.0 (fma (* -1.0 (* sinTheta_O sinTheta_i)) v (* -1.0 v))) t_1)))
    (fma
     cosTheta_O
     (* (/ (* (* cosTheta_O (* cosTheta_i cosTheta_i)) t_0) t_1) 0.5)
     (/ (* (* cosTheta_O cosTheta_i) t_0) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = expf((((0.6931f + logf(0.5f)) + (logf(v) * -1.0f)) - (1.0f / v))) / expf(((sinTheta_O * sinTheta_i) / v));
	float t_1 = -1.0f * ((-1.0f * v) * v);
	return fmaf(expf(0.6931f), expf((logf((0.5f * (1.0f / v))) - ((-1.0f * fmaf((-1.0f * (sinTheta_O * sinTheta_i)), v, (-1.0f * v))) / t_1))), fmaf(cosTheta_O, ((((cosTheta_O * (cosTheta_i * cosTheta_i)) * t_0) / t_1) * 0.5f), (((cosTheta_O * cosTheta_i) * t_0) / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(exp(Float32(Float32(Float32(Float32(0.6931) + log(Float32(0.5))) + Float32(log(v) * Float32(-1.0))) - Float32(Float32(1.0) / v))) / exp(Float32(Float32(sinTheta_O * sinTheta_i) / v)))
	t_1 = Float32(Float32(-1.0) * Float32(Float32(Float32(-1.0) * v) * v))
	return fma(exp(Float32(0.6931)), exp(Float32(log(Float32(Float32(0.5) * Float32(Float32(1.0) / v))) - Float32(Float32(Float32(-1.0) * fma(Float32(Float32(-1.0) * Float32(sinTheta_O * sinTheta_i)), v, Float32(Float32(-1.0) * v))) / t_1))), fma(cosTheta_O, Float32(Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i * cosTheta_i)) * t_0) / t_1) * Float32(0.5)), Float32(Float32(Float32(cosTheta_O * cosTheta_i) * t_0) / v)))
end

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} + cosTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)}{{v}^{2}} + \frac{cosTheta\_i \cdot e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v}\right)} \]

5. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{0.6931}, e^{\log \left(0.5 \cdot \frac{1}{v}\right) - \frac{\mathsf{fma}\left(-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v, \left(-v\right) \cdot 1\right)}{\left(-v\right) \cdot v}}, \mathsf{fma}\left(cosTheta\_O, \frac{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v \cdot v} \cdot 0.5, \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v}\right)\right)} \]

6. Final simplification 50.6%

   \[\leadsto \mathsf{fma}\left(e^{0.6931}, e^{\log \left(0.5 \cdot \frac{1}{v}\right) - \frac{-1 \cdot \mathsf{fma}\left(-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v, -1 \cdot v\right)}{-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)}}, \mathsf{fma}\left(cosTheta\_O, \frac{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)} \cdot 0.5, \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v}\right)\right) \]
7. Add Preprocessing

Alternative 3: 55.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\\ t_1 := -1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)\\ \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot t\_0, \frac{\left(-1 \cdot e^{0.6931}\right) \cdot e^{\log \left(\frac{0.5}{v}\right) - \frac{-1 \cdot \mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{t\_1}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{t\_1} \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (/
          (/
           (* (exp 0.6931) (* (exp (log 0.5)) (exp (* -1.0 (log v)))))
           (exp (/ 1.0 v)))
          (exp (/ (* sinTheta_O sinTheta_i) v))))
        (t_1 (* -1.0 (* (* -1.0 v) v))))
   (*
    (* cosTheta_O cosTheta_O)
    (fma
     -1.0
     (/
      (fma
       -1.0
       (* (/ cosTheta_i v) t_0)
       (/
        (*
         (* -1.0 (exp 0.6931))
         (exp
          (-
           (log (/ 0.5 v))
           (/
            (* -1.0 (fma -1.0 v (* -1.0 (* sinTheta_O (* sinTheta_i v)))))
            t_1))))
        cosTheta_O))
      cosTheta_O)
     (* 0.5 (* (/ (* cosTheta_i cosTheta_i) t_1) t_0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = ((expf(0.6931f) * (expf(logf(0.5f)) * expf((-1.0f * logf(v))))) / expf((1.0f / v))) / expf(((sinTheta_O * sinTheta_i) / v));
	float t_1 = -1.0f * ((-1.0f * v) * v);
	return (cosTheta_O * cosTheta_O) * fmaf(-1.0f, (fmaf(-1.0f, ((cosTheta_i / v) * t_0), (((-1.0f * expf(0.6931f)) * expf((logf((0.5f / v)) - ((-1.0f * fmaf(-1.0f, v, (-1.0f * (sinTheta_O * (sinTheta_i * v))))) / t_1)))) / cosTheta_O)) / cosTheta_O), (0.5f * (((cosTheta_i * cosTheta_i) / t_1) * t_0)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(exp(Float32(0.6931)) * Float32(exp(log(Float32(0.5))) * exp(Float32(Float32(-1.0) * log(v))))) / exp(Float32(Float32(1.0) / v))) / exp(Float32(Float32(sinTheta_O * sinTheta_i) / v)))
	t_1 = Float32(Float32(-1.0) * Float32(Float32(Float32(-1.0) * v) * v))
	return Float32(Float32(cosTheta_O * cosTheta_O) * fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(Float32(cosTheta_i / v) * t_0), Float32(Float32(Float32(Float32(-1.0) * exp(Float32(0.6931))) * exp(Float32(log(Float32(Float32(0.5) / v)) - Float32(Float32(Float32(-1.0) * fma(Float32(-1.0), v, Float32(Float32(-1.0) * Float32(sinTheta_O * Float32(sinTheta_i * v))))) / t_1)))) / cosTheta_O)) / cosTheta_O), Float32(Float32(0.5) * Float32(Float32(Float32(cosTheta_i * cosTheta_i) / t_1) * t_0))))
end

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} + cosTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)}{{v}^{2}} + \frac{cosTheta\_i \cdot e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v}\right)} \]

5. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{0.6931}, e^{\log \left(0.5 \cdot \frac{1}{v}\right) - \frac{\mathsf{fma}\left(-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v, \left(-v\right) \cdot 1\right)}{\left(-v\right) \cdot v}}, \mathsf{fma}\left(cosTheta\_O, \frac{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v \cdot v} \cdot 0.5, \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v}\right)\right)} \]

6. Taylor expanded in cosTheta_O around -inf

   \[\leadsto {cosTheta\_O}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{cosTheta\_i \cdot e^{\left(\frac{6931}{10000} + \left(\log \frac{1}{2} + -1 \cdot \log v\right)\right) - \frac{1}{v}}}{v \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} + -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{-1 \cdot v + -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)}{{v}^{2}}}}{cosTheta\_O}}{cosTheta\_O} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2} \cdot e^{\left(\frac{6931}{10000} + \left(\log \frac{1}{2} + -1 \cdot \log v\right)\right) - \frac{1}{v}}}{{v}^{2} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)} \]

8. Applied rewrites 49.7%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right)} \]

9. Final simplification 49.7%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, \frac{\left(-1 \cdot e^{0.6931}\right) \cdot e^{\log \left(\frac{0.5}{v}\right) - \frac{-1 \cdot \mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]
10. Add Preprocessing

Alternative 4: 41.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)\\ t_1 := e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)\\ t_2 := e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{t\_1}{1 + \frac{\mathsf{fma}\left(0.5, v, t\_0\right)}{t\_0 \cdot v}}}{t\_2}, \frac{\left(-1 \cdot e^{0.6931}\right) \cdot e^{\log \left(\frac{0.5}{v}\right) - \frac{-1 \cdot \mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{t\_0}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{t\_0} \cdot \frac{\frac{t\_1}{e^{\frac{1}{v}}}}{t\_2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* -1.0 (* (* -1.0 v) v)))
        (t_1 (* (exp 0.6931) (* (exp (log 0.5)) (exp (* -1.0 (log v))))))
        (t_2 (exp (/ (* sinTheta_O sinTheta_i) v))))
   (*
    (* cosTheta_O cosTheta_O)
    (fma
     -1.0
     (/
      (fma
       -1.0
       (*
        (/ cosTheta_i v)
        (/ (/ t_1 (+ 1.0 (/ (fma 0.5 v t_0) (* t_0 v)))) t_2))
       (/
        (*
         (* -1.0 (exp 0.6931))
         (exp
          (-
           (log (/ 0.5 v))
           (/
            (* -1.0 (fma -1.0 v (* -1.0 (* sinTheta_O (* sinTheta_i v)))))
            t_0))))
        cosTheta_O))
      cosTheta_O)
     (*
      0.5
      (*
       (/ (* cosTheta_i cosTheta_i) t_0)
       (/ (/ t_1 (exp (/ 1.0 v))) t_2)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = -1.0f * ((-1.0f * v) * v);
	float t_1 = expf(0.6931f) * (expf(logf(0.5f)) * expf((-1.0f * logf(v))));
	float t_2 = expf(((sinTheta_O * sinTheta_i) / v));
	return (cosTheta_O * cosTheta_O) * fmaf(-1.0f, (fmaf(-1.0f, ((cosTheta_i / v) * ((t_1 / (1.0f + (fmaf(0.5f, v, t_0) / (t_0 * v)))) / t_2)), (((-1.0f * expf(0.6931f)) * expf((logf((0.5f / v)) - ((-1.0f * fmaf(-1.0f, v, (-1.0f * (sinTheta_O * (sinTheta_i * v))))) / t_0)))) / cosTheta_O)) / cosTheta_O), (0.5f * (((cosTheta_i * cosTheta_i) / t_0) * ((t_1 / expf((1.0f / v))) / t_2))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(-1.0) * Float32(Float32(Float32(-1.0) * v) * v))
	t_1 = Float32(exp(Float32(0.6931)) * Float32(exp(log(Float32(0.5))) * exp(Float32(Float32(-1.0) * log(v)))))
	t_2 = exp(Float32(Float32(sinTheta_O * sinTheta_i) / v))
	return Float32(Float32(cosTheta_O * cosTheta_O) * fma(Float32(-1.0), Float32(fma(Float32(-1.0), Float32(Float32(cosTheta_i / v) * Float32(Float32(t_1 / Float32(Float32(1.0) + Float32(fma(Float32(0.5), v, t_0) / Float32(t_0 * v)))) / t_2)), Float32(Float32(Float32(Float32(-1.0) * exp(Float32(0.6931))) * exp(Float32(log(Float32(Float32(0.5) / v)) - Float32(Float32(Float32(-1.0) * fma(Float32(-1.0), v, Float32(Float32(-1.0) * Float32(sinTheta_O * Float32(sinTheta_i * v))))) / t_0)))) / cosTheta_O)) / cosTheta_O), Float32(Float32(0.5) * Float32(Float32(Float32(cosTheta_i * cosTheta_i) / t_0) * Float32(Float32(t_1 / exp(Float32(Float32(1.0) / v))) / t_2)))))
end

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} + cosTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right)}{{v}^{2}} + \frac{cosTheta\_i \cdot e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v}\right)} \]

5. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{0.6931}, e^{\log \left(0.5 \cdot \frac{1}{v}\right) - \frac{\mathsf{fma}\left(-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), v, \left(-v\right) \cdot 1\right)}{\left(-v\right) \cdot v}}, \mathsf{fma}\left(cosTheta\_O, \frac{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v \cdot v} \cdot 0.5, \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{e^{\left(\left(0.6931 + \log 0.5\right) + \log v \cdot -1\right) - \frac{1}{v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v}\right)\right)} \]

6. Taylor expanded in cosTheta_O around -inf

   \[\leadsto {cosTheta\_O}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{cosTheta\_i \cdot e^{\left(\frac{6931}{10000} + \left(\log \frac{1}{2} + -1 \cdot \log v\right)\right) - \frac{1}{v}}}{v \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} + -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{-1 \cdot v + -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)}{{v}^{2}}}}{cosTheta\_O}}{cosTheta\_O} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2} \cdot e^{\left(\frac{6931}{10000} + \left(\log \frac{1}{2} + -1 \cdot \log v\right)\right) - \frac{1}{v}}}{{v}^{2} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)} \]

8. Applied rewrites 49.7%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right)} \]

9. Taylor expanded in v around inf

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \left(\frac{\frac{1}{2}}{{v}^{2}} + \frac{1}{v}\right)}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]
10. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \left(\frac{\frac{1}{2}}{{v}^{2}} + \frac{1}{v}\right)}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    2. frac-add N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\frac{1}{2} \cdot v + {v}^{2} \cdot 1}{{v}^{2} \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    3. pow2 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\frac{1}{2} \cdot v + {v}^{2} \cdot 1}{\left(v \cdot v\right) \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    4. unpow3 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\frac{1}{2} \cdot v + {v}^{2} \cdot 1}{{v}^{3}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    5. lower-/.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\frac{1}{2} \cdot v + {v}^{2} \cdot 1}{{v}^{3}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    6. lower-fma.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, {v}^{2} \cdot 1\right)}{{v}^{3}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    7. lower-*.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, {v}^{2} \cdot 1\right)}{{v}^{3}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    8. pow2 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, \left(v \cdot v\right) \cdot 1\right)}{{v}^{3}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    9. lift-*.f32 N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, \left(v \cdot v\right) \cdot 1\right)}{{v}^{3}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    10. unpow3 N/A

        \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, \left(v \cdot v\right) \cdot 1\right)}{\left(v \cdot v\right) \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    11. pow2 N/A

        \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, \left(v \cdot v\right) \cdot 1\right)}{{v}^{2} \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    12. lower-*.f32 N/A

        \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, \left(v \cdot v\right) \cdot 1\right)}{{v}^{2} \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    13. pow2 N/A

        \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(\frac{1}{2}, v, \left(v \cdot v\right) \cdot 1\right)}{\left(v \cdot v\right) \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{\frac{6931}{10000}} \cdot e^{\log \left(\frac{\frac{1}{2}}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, \frac{1}{2} \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{\frac{6931}{10000}} \cdot \left(e^{\log \frac{1}{2}} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

    14. lift-*.f32 38.0

        \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(0.5, v, \left(v \cdot v\right) \cdot 1\right)}{\left(v \cdot v\right) \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

11. Applied rewrites 38.0%

    \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(0.5, v, \left(v \cdot v\right) \cdot 1\right)}{\left(v \cdot v\right) \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, -1 \cdot \frac{e^{0.6931} \cdot e^{\log \left(\frac{0.5}{v}\right) - -1 \cdot \frac{\mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{v \cdot v}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]

12. Final simplification 38.0%

    \[\leadsto \left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{cosTheta\_i}{v} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{1 + \frac{\mathsf{fma}\left(0.5, v, -1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)\right)}{\left(-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)\right) \cdot v}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}, \frac{\left(-1 \cdot e^{0.6931}\right) \cdot e^{\log \left(\frac{0.5}{v}\right) - \frac{-1 \cdot \mathsf{fma}\left(-1, v, -1 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)\right)}{-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)}}}{cosTheta\_O}\right)}{cosTheta\_O}, 0.5 \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_i}{-1 \cdot \left(\left(-1 \cdot v\right) \cdot v\right)} \cdot \frac{\frac{e^{0.6931} \cdot \left(e^{\log 0.5} \cdot e^{-1 \cdot \log v}\right)}{e^{\frac{1}{v}}}}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))