HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.8%

Time: 10.2s

Alternatives: 8

Speedup: 2.1×

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: 99.6% 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: 99.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0.6931 - \frac{1}{v}}}{v} \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (exp (- 0.6931 (/ 1.0 v))) v) 0.5))

C

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

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((0.6931e0 - (1.0e0 / v))) / v) * 0.5e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp((single(0.6931) - (single(1.0) / v))) / v) * single(0.5);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower--.f32 N/A

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

   9. div-add-rev N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}} \]

   12. lower-fma.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.9%

   \[\leadsto \frac{e^{0.6931 - \frac{1}{v}}}{v} \cdot \color{blue}{0.5} \]
9. Add Preprocessing

Alternative 2: 18.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.0000000718782596 \cdot 10^{-36}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.0000000718782596e-36)
   (exp (* cosTheta_O (/ cosTheta_i v)))
   (exp (/ (* sinTheta_O sinTheta_i) (- v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.0000000718782596e-36f) {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	} else {
		tmp = expf(((sinTheta_O * sinTheta_i) / -v));
	}
	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(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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.0000000718782596e-36) then
        tmp = exp((costheta_o * (costheta_i / v)))
    else
        tmp = exp(((sintheta_o * sintheta_i) / -v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.0000000718782596e-36))
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	else
		tmp = exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.0000000718782596e-36))
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	else
		tmp = exp(((sinTheta_O * sinTheta_i) / -v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000007e-36

   1. Initial program 99.3%

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

   3. Taylor expanded in cosTheta_i around 0

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   4. Step-by-step derivation

      1. lower--.f32 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 99.3

         \[\leadsto e^{\left(0.6931 - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto e^{\color{blue}{\left(0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-log.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-log.f32 99.8

         \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(2 \cdot v\right)}, -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v}\right)}} \]

   8. Taylor expanded in sinTheta_i around 0

      \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, \frac{6931}{10000} - \color{blue}{\frac{1}{v}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \color{blue}{\frac{1}{v}}\right)} \]

   11. Taylor expanded in cosTheta_i around inf

       \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

       2. lower-*.f32 N/A

          \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

       3. lower-/.f32 13.1

          \[\leadsto e^{cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i}{v}}} \]

   13. Applied rewrites 13.1%

       \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

   if 2.00000007e-36 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around 0

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   4. Step-by-step derivation

      1. lower--.f32 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 100.0

         \[\leadsto e^{\left(0.6931 - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto e^{\color{blue}{\left(0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-log.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-log.f32 100.0

         \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(2 \cdot v\right)}, -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v}\right)}} \]

   8. Taylor expanded in sinTheta_i around 0

      \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, \frac{6931}{10000} - \color{blue}{\frac{1}{v}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \color{blue}{\frac{1}{v}}\right)} \]

   11. Taylor expanded in sinTheta_i around inf

       \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

       2. lower-neg.f32 N/A

          \[\leadsto e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

       3. lower-/.f32 N/A

          \[\leadsto e^{-\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

       4. lower-*.f32 32.3

          \[\leadsto e^{-\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   13. Applied rewrites 32.3%

       \[\leadsto e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.0000000718782596 \cdot 10^{-36}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-1}{v}}}{v} \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (exp (/ -1.0 v)) v) 0.5))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf((-1.0f / v)) / v) * 0.5f;
}

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(((-1.0e0) / v)) / v) * 0.5e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(-1.0) / v)) / v) * Float32(0.5))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp((single(-1.0) / v)) / v) * single(0.5);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower--.f32 N/A

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

   9. div-add-rev N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}} \]

   12. lower-fma.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.9%

   \[\leadsto \frac{e^{0.6931 - \frac{1}{v}}}{v} \cdot \color{blue}{0.5} \]

9. Taylor expanded in v around 0

   \[\leadsto \frac{e^{\frac{-1}{v}}}{v} \cdot \frac{1}{2} \]
10. Step-by-step derivation

11. Applied rewrites 98.8%

    \[\leadsto \frac{e^{\frac{-1}{v}}}{v} \cdot 0.5 \]
12. Add Preprocessing

Alternative 4: 97.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ (- (* cosTheta_O cosTheta_i) (fma sinTheta_O sinTheta_i 1.0)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((cosTheta_O * cosTheta_i) - fmaf(sinTheta_O, sinTheta_i, 1.0f)) / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - fma(sinTheta_O, sinTheta_i, Float32(1.0))) / v))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
4. Step-by-step derivation

   1. lower--.f32 N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 99.5

      \[\leadsto e^{\left(0.6931 - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

5. Applied rewrites 99.5%

   \[\leadsto e^{\color{blue}{\left(0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. rem-log-exp N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-log.f32 99.9

      \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(2 \cdot v\right)}, -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} \]

7. Applied rewrites 99.9%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v}\right)}} \]

8. Taylor expanded in sinTheta_i around 0

   \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, \frac{6931}{10000} - \color{blue}{\frac{1}{v}}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

    \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \color{blue}{\frac{1}{v}}\right)} \]

11. Taylor expanded in v around 0

    \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f32 98.7

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

13. Applied rewrites 98.7%

    \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]
14. Add Preprocessing

Alternative 5: 13.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (* cosTheta_O (/ cosTheta_i v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((cosTheta_O * (cosTheta_i / 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_o * (costheta_i / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(cosTheta_O * Float32(cosTheta_i / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((cosTheta_O * (cosTheta_i / v)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
4. Step-by-step derivation

   1. lower--.f32 N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 99.5

      \[\leadsto e^{\left(0.6931 - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

5. Applied rewrites 99.5%

   \[\leadsto e^{\color{blue}{\left(0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. rem-log-exp N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-log.f32 99.9

      \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(2 \cdot v\right)}, -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} \]

7. Applied rewrites 99.9%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v}\right)}} \]

8. Taylor expanded in sinTheta_i around 0

   \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, \frac{6931}{10000} - \color{blue}{\frac{1}{v}}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

    \[\leadsto e^{\mathsf{fma}\left(\log \left(2 \cdot v\right), -1, 0.6931 - \color{blue}{\frac{1}{v}}\right)} \]

11. Taylor expanded in cosTheta_i around inf

    \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
12. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

    2. lower-*.f32 N/A

       \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

    3. lower-/.f32 11.9

       \[\leadsto e^{cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i}{v}}} \]

13. Applied rewrites 11.9%

    \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
14. Add Preprocessing

Alternative 6: 4.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp 0.6931)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf(0.6931f);
}

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 = (0.5e0 / v) * exp(0.6931e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(0.6931)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp(single(0.6931));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \left(\log \frac{1}{2} + \log \left(\frac{1}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \frac{1}{2} + \log \left(\frac{1}{v}\right)\right) + \frac{6931}{10000}}} \]

   2. exp-sum N/A

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

   3. log-rec N/A

      \[\leadsto e^{\log \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\log v\right)\right)}} \cdot e^{\frac{6931}{10000}} \]

   4. mul-1-neg N/A

      \[\leadsto e^{\log \frac{1}{2} + \color{blue}{-1 \cdot \log v}} \cdot e^{\frac{6931}{10000}} \]

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

      \[\leadsto e^{\color{blue}{\log \frac{1}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \log v}} \cdot e^{\frac{6931}{10000}} \]

   6. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\log \frac{1}{2}}}{e^{\left(\mathsf{neg}\left(-1\right)\right) \cdot \log v}}} \cdot e^{\frac{6931}{10000}} \]

   7. rem-exp-log N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{e^{\left(\mathsf{neg}\left(-1\right)\right) \cdot \log v}} \cdot e^{\frac{6931}{10000}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{e^{\color{blue}{1} \cdot \log v}} \cdot e^{\frac{6931}{10000}} \]

   9. *-lft-identity N/A

      \[\leadsto \frac{\frac{1}{2}}{e^{\color{blue}{\log v}}} \cdot e^{\frac{6931}{10000}} \]

   10. rem-exp-log N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{v}} \cdot e^{\frac{6931}{10000}} \]

   11. rem-exp-log N/A

       \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)}} \cdot e^{\frac{6931}{10000}} \]

   12. lower-*.f32 N/A

       \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000}}} \]

   13. rem-exp-log N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000}} \]

   14. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000}} \]

   15. lower-exp.f32 4.6

       \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931}} \]

5. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931}} \]
6. Add Preprocessing

Alternative 7: 4.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0.6931}}{v + v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp 0.6931) (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(0.6931f) / (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(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(0.6931e0) / (v + v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(0.6931)) / Float32(v + v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(single(0.6931)) / (v + v);
end

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. metadata-eval N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \left(\color{blue}{0} - \log \left(2 \cdot v\right)\right)} \]

   7. associate-+r- N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0\right) - \log \left(2 \cdot v\right)}} \]

   8. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0}}{e^{\log \left(2 \cdot v\right)}}} \]

   9. rem-exp-log N/A

      \[\leadsto \frac{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0}}{\color{blue}{2 \cdot v}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0}}{2 \cdot v}} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right) + 0}}{2 \cdot v}} \]

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\color{blue}{\frac{6931}{10000}}}}{2 \cdot v} \]
6. Step-by-step derivation

7. Applied rewrites 4.6%

   \[\leadsto \frac{e^{\color{blue}{0.6931}}}{2 \cdot v} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{6931}{10000}}}{\color{blue}{2 \cdot v}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{e^{\frac{6931}{10000}}}{\color{blue}{v + v}} \]

   3. lower-+.f32 4.6

      \[\leadsto \frac{e^{0.6931}}{\color{blue}{v + v}} \]

9. Applied rewrites 4.6%

   \[\leadsto \frac{e^{0.6931}}{\color{blue}{v + v}} \]
10. Add Preprocessing

Alternative 8: 3.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0.6931}}{0} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp 0.6931) 0.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(0.6931f) / 0.0f;
}

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(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(0.6931e0) / 0.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(0.6931)) / Float32(0.0))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(single(0.6931)) / single(0.0);
end

Derivation

1. Initial program 99.5%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. metadata-eval N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \left(\color{blue}{0} - \log \left(2 \cdot v\right)\right)} \]

   7. associate-+r- N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0\right) - \log \left(2 \cdot v\right)}} \]

   8. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0}}{e^{\log \left(2 \cdot v\right)}}} \]

   9. rem-exp-log N/A

      \[\leadsto \frac{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0}}{\color{blue}{2 \cdot v}} \]

   10. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + 0}}{2 \cdot v}} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right) + 0}}{2 \cdot v}} \]

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\color{blue}{\frac{6931}{10000}}}}{2 \cdot v} \]
6. Step-by-step derivation

7. Applied rewrites 4.6%

   \[\leadsto \frac{e^{\color{blue}{0.6931}}}{2 \cdot v} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{\frac{6931}{10000}}}{\color{blue}{2 \cdot v}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{e^{\frac{6931}{10000}}}{\color{blue}{v + v}} \]

   3. lower-+.f32 4.6

      \[\leadsto \frac{e^{0.6931}}{\color{blue}{v + v}} \]

9. Applied rewrites 4.6%

   \[\leadsto \frac{e^{0.6931}}{\color{blue}{v + v}} \]
10. Step-by-step derivation

    1. lift-+.f32 N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\color{blue}{v + v}} \]

    2. flip-+ N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\color{blue}{\frac{v \cdot v - v \cdot v}{v - v}}} \]

    3. +-inverses N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{\color{blue}{0}}{v - v}} \]

    4. +-inverses N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{0}{\color{blue}{0}}} \]

    5. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{\color{blue}{0 - 0}}{0}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{\color{blue}{{0}^{3}} - 0}{0}} \]

    7. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - \color{blue}{{0}^{3}}}{0}} \]

    8. mul0-rgt N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\color{blue}{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) \cdot 0\right)}}^{3}}{0}} \]

    9. sub-div N/A

       \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + \frac{6931}{10000}\right) \cdot 0\right)}^{3}}{0}} \]

    10. *-commutative N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\left(\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i} - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + \frac{6931}{10000}\right) \cdot 0\right)}^{3}}{0}} \]

    11. *-commutative N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} - \frac{1}{v}\right) + \frac{6931}{10000}\right) \cdot 0\right)}^{3}}{0}} \]

    12. sub-div N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} + \frac{6931}{10000}\right) \cdot 0\right)}^{3}}{0}} \]

    13. associate--r+ N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}}{v} + \frac{6931}{10000}\right) \cdot 0\right)}^{3}}{0}} \]

    14. +-commutative N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\color{blue}{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}{v}\right)} \cdot 0\right)}^{3}}{0}} \]

    15. metadata-eval N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}{v}\right) \cdot 0\right)}^{3}}{\color{blue}{0 + 0}}} \]

    16. metadata-eval N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}{v}\right) \cdot 0\right)}^{3}}{\color{blue}{0 \cdot 0} + 0}} \]

    17. metadata-eval N/A

        \[\leadsto \frac{e^{\frac{6931}{10000}}}{\frac{{0}^{3} - {\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}{v}\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \color{blue}{\left(0 + 0\right)}}} \]

11. Applied rewrites 3.2%

    \[\leadsto \color{blue}{\frac{e^{0.6931}}{0}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024353 
(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))))))