HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.6%

Time: 14.2s

Alternatives: 15

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / (v * (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. lower-exp.f32 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 98.5

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

5. Simplified 98.5%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = ((costheta_i * costheta_o) / v) / (v * (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. lower-exp.f32 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 98.5

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

5. Simplified 98.5%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

   \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \]
10. Add Preprocessing

Alternative 3: 81.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.26499998569488525:\\ \;\;\;\;\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.26499998569488525)
   (/ (/ (* cosTheta_i cosTheta_O) v) (* v (+ (exp (/ 1.0 v)) -1.0)))
   (/
    (* cosTheta_i (/ cosTheta_O v))
    (*
     v
     (/
      (+
       (+ 2.0 (/ 0.3333333333333333 (* v v)))
       (/
        (+ 0.016666666666666666 (/ 0.0003968253968253968 (* v v)))
        (pow v 4.0)))
      v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.26499998569488525f) {
		tmp = ((cosTheta_i * cosTheta_O) / v) / (v * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = (cosTheta_i * (cosTheta_O / v)) / (v * (((2.0f + (0.3333333333333333f / (v * v))) + ((0.016666666666666666f + (0.0003968253968253968f / (v * v))) / powf(v, 4.0f))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 (v <= 0.26499998569488525e0) then
        tmp = ((costheta_i * costheta_o) / v) / (v * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = (costheta_i * (costheta_o / v)) / (v * (((2.0e0 + (0.3333333333333333e0 / (v * v))) + ((0.016666666666666666e0 + (0.0003968253968253968e0 / (v * v))) / (v ** 4.0e0))) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.26499998569488525))
		tmp = Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) / Float32(v * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) + Float32(Float32(Float32(0.016666666666666666) + Float32(Float32(0.0003968253968253968) / Float32(v * v))) / (v ^ Float32(4.0)))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.26499998569488525))
		tmp = ((cosTheta_i * cosTheta_O) / v) / (v * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = (cosTheta_i * (cosTheta_O / v)) / (v * (((single(2.0) + (single(0.3333333333333333) / (v * v))) + ((single(0.016666666666666666) + (single(0.0003968253968253968) / (v * v))) / (v ^ single(4.0)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.264999986

   1. Initial program 98.1%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Simplified 98.2%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 76.7%

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

   if 0.264999986 < v

   1. Initial program 98.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Simplified 98.7%

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

   9. Taylor expanded in v around -inf

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}}} \]

       3. sub-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

       5. distribute-neg-out N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       6. remove-double-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

       7. lower-/.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

   11. Simplified 82.5%

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}} \]

   12. Taylor expanded in cosTheta_i around 0

       \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60} + \frac{\frac{1}{2520}}{v \cdot v}}{{v}^{4}}}{v}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60} + \frac{\frac{1}{2520}}{v \cdot v}}{{v}^{4}}}{v}} \]

       2. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60} + \frac{\frac{1}{2520}}{v \cdot v}}{{v}^{4}}}{v}} \]

       3. lower-*.f32 N/A

          \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60} + \frac{\frac{1}{2520}}{v \cdot v}}{{v}^{4}}}{v}} \]

       4. lower-/.f32 82.5

          \[\leadsto \frac{cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}} \]

   14. Simplified 82.5%

       \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.26499998569488525:\\ \;\;\;\;\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;v \leq 0.26499998569488525:\\ \;\;\;\;\frac{t\_0}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{\mathsf{fma}\left(v, v \cdot 0.016666666666666666, 0.0003968253968253968\right)}{{v}^{6}}}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<= v 0.26499998569488525)
     (/ t_0 (* v (+ (exp (/ 1.0 v)) -1.0)))
     (/
      t_0
      (*
       v
       (/
        (+
         (+ 2.0 (/ 0.3333333333333333 (* v v)))
         (/
          (fma v (* v 0.016666666666666666) 0.0003968253968253968)
          (pow v 6.0)))
        v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if (v <= 0.26499998569488525f) {
		tmp = t_0 / (v * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = t_0 / (v * (((2.0f + (0.3333333333333333f / (v * v))) + (fmaf(v, (v * 0.016666666666666666f), 0.0003968253968253968f) / powf(v, 6.0f))) / v));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (v <= Float32(0.26499998569488525))
		tmp = Float32(t_0 / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(t_0 / Float32(v * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) + Float32(fma(v, Float32(v * Float32(0.016666666666666666)), Float32(0.0003968253968253968)) / (v ^ Float32(6.0)))) / v)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if v < 0.264999986

   1. Initial program 98.1%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Simplified 98.2%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 76.7%

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

   if 0.264999986 < v

   1. Initial program 98.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Simplified 98.7%

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

   9. Taylor expanded in v around -inf

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}}} \]

       3. sub-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

       5. distribute-neg-out N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       6. remove-double-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

       7. lower-/.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

   11. Simplified 82.5%

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}} \]

   12. Taylor expanded in v around 0

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \color{blue}{\frac{\frac{1}{2520} + \frac{1}{60} \cdot {v}^{2}}{{v}^{6}}}}{v}} \]
   13. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \color{blue}{\frac{\frac{1}{2520} + \frac{1}{60} \cdot {v}^{2}}{{v}^{6}}}}{v}} \]

       2. +-commutative N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\color{blue}{\frac{1}{60} \cdot {v}^{2} + \frac{1}{2520}}}{{v}^{6}}}{v}} \]

       3. *-commutative N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\color{blue}{{v}^{2} \cdot \frac{1}{60}} + \frac{1}{2520}}{{v}^{6}}}{v}} \]

       4. unpow2 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\color{blue}{\left(v \cdot v\right)} \cdot \frac{1}{60} + \frac{1}{2520}}{{v}^{6}}}{v}} \]

       5. associate-*l* N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\color{blue}{v \cdot \left(v \cdot \frac{1}{60}\right)} + \frac{1}{2520}}{{v}^{6}}}{v}} \]

       6. lower-fma.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\color{blue}{\mathsf{fma}\left(v, v \cdot \frac{1}{60}, \frac{1}{2520}\right)}}{{v}^{6}}}{v}} \]

       7. lower-*.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\mathsf{fma}\left(v, \color{blue}{v \cdot \frac{1}{60}}, \frac{1}{2520}\right)}{{v}^{6}}}{v}} \]

       8. lower-pow.f32 82.5

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{\mathsf{fma}\left(v, v \cdot 0.016666666666666666, 0.0003968253968253968\right)}{\color{blue}{{v}^{6}}}}{v}} \]

   14. Simplified 82.5%

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \color{blue}{\frac{\mathsf{fma}\left(v, v \cdot 0.016666666666666666, 0.0003968253968253968\right)}{{v}^{6}}}}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.26499998569488525:\\ \;\;\;\;\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{\mathsf{fma}\left(v, v \cdot 0.016666666666666666, 0.0003968253968253968\right)}{{v}^{6}}}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;v \leq 0.26499998569488525:\\ \;\;\;\;\frac{t\_0}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<= v 0.26499998569488525)
     (/ t_0 (* v (+ (exp (/ 1.0 v)) -1.0)))
     (/
      t_0
      (*
       v
       (/
        (+
         (+ 2.0 (/ 0.3333333333333333 (* v v)))
         (/
          (+ 0.016666666666666666 (/ 0.0003968253968253968 (* v v)))
          (* (* v v) (* v v))))
        v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if (v <= 0.26499998569488525f) {
		tmp = t_0 / (v * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = t_0 / (v * (((2.0f + (0.3333333333333333f / (v * v))) + ((0.016666666666666666f + (0.0003968253968253968f / (v * v))) / ((v * v) * (v * v)))) / v));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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) :: t_0
    real(4) :: tmp
    t_0 = (costheta_i * costheta_o) / v
    if (v <= 0.26499998569488525e0) then
        tmp = t_0 / (v * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = t_0 / (v * (((2.0e0 + (0.3333333333333333e0 / (v * v))) + ((0.016666666666666666e0 + (0.0003968253968253968e0 / (v * v))) / ((v * v) * (v * v)))) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (v <= Float32(0.26499998569488525))
		tmp = Float32(t_0 / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(t_0 / Float32(v * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) + Float32(Float32(Float32(0.016666666666666666) + Float32(Float32(0.0003968253968253968) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v)))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_i * cosTheta_O) / v;
	tmp = single(0.0);
	if (v <= single(0.26499998569488525))
		tmp = t_0 / (v * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = t_0 / (v * (((single(2.0) + (single(0.3333333333333333) / (v * v))) + ((single(0.016666666666666666) + (single(0.0003968253968253968) / (v * v))) / ((v * v) * (v * v)))) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.264999986

   1. Initial program 98.1%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Simplified 98.2%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 76.7%

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

   if 0.264999986 < v

   1. Initial program 98.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-exp.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Simplified 98.7%

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

   9. Taylor expanded in v around -inf

      \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}}} \]

       3. sub-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

       5. distribute-neg-out N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       6. remove-double-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

       7. lower-/.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

   11. Simplified 82.5%

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}} \]

   12. Taylor expanded in v around -inf

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}{v}}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}}{v}} \]

       3. sub-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

       5. distribute-neg-out N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

       6. remove-double-neg N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

       7. lower-/.f32 N/A

          \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

   14. Simplified 82.5%

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.26499998569488525:\\ \;\;\;\;\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_i cosTheta_O) v)
  (*
   v
   (/
    (+
     (+ 2.0 (/ 0.3333333333333333 (* v v)))
     (/
      (+ 0.016666666666666666 (/ 0.0003968253968253968 (* v v)))
      (* (* v v) (* v v))))
    v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) / (v * (((2.0f + (0.3333333333333333f / (v * v))) + ((0.016666666666666666f + (0.0003968253968253968f / (v * v))) / ((v * v) * (v * v)))) / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = ((costheta_i * costheta_o) / v) / (v * (((2.0e0 + (0.3333333333333333e0 / (v * v))) + ((0.016666666666666666e0 + (0.0003968253968253968e0 / (v * v))) / ((v * v) * (v * v)))) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) / Float32(v * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) + Float32(Float32(Float32(0.016666666666666666) + Float32(Float32(0.0003968253968253968) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v)))) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) / (v * (((single(2.0) + (single(0.3333333333333333) / (v * v))) + ((single(0.016666666666666666) + (single(0.0003968253968253968) / (v * v))) / ((v * v) * (v * v)))) / v));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. lower-exp.f32 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 98.5

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

5. Simplified 98.5%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Simplified 98.5%

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

9. Taylor expanded in v around -inf

   \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)}} \]

    2. distribute-frac-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}}} \]

    3. sub-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

    5. distribute-neg-out N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

11. Simplified 76.6%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}} \]

12. Taylor expanded in v around -inf

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}{v}}} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}}{v}} \]

    3. sub-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

    5. distribute-neg-out N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

14. Simplified 76.6%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)} + \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}{v}}} \]

15. Final simplification 76.6%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}} \]
16. Add Preprocessing

Alternative 7: 70.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_i cosTheta_O) v)
  (*
   v
   (/
    (+
     (+ 2.0 (/ 0.3333333333333333 (* v v)))
     (/ 0.016666666666666666 (* (* v v) (* v v))))
    v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) / (v * (((2.0f + (0.3333333333333333f / (v * v))) + (0.016666666666666666f / ((v * v) * (v * v)))) / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = ((costheta_i * costheta_o) / v) / (v * (((2.0e0 + (0.3333333333333333e0 / (v * v))) + (0.016666666666666666e0 / ((v * v) * (v * v)))) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) / Float32(v * Float32(Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) + Float32(Float32(0.016666666666666666) / Float32(Float32(v * v) * Float32(v * v)))) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) / (v * (((single(2.0) + (single(0.3333333333333333) / (v * v))) + (single(0.016666666666666666) / ((v * v) * (v * v)))) / v));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. lower-exp.f32 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 98.5

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

5. Simplified 98.5%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Simplified 98.5%

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

9. Taylor expanded in v around -inf

   \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)}} \]

    2. distribute-frac-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}}} \]

    3. sub-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v}} \]

    5. distribute-neg-out N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}{v}} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{60} + \frac{1}{2520} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}{v}}} \]

11. Simplified 76.6%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666 + \frac{0.0003968253968253968}{v \cdot v}}{{v}^{4}}}{v}}} \]

12. Taylor expanded in v around inf

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \color{blue}{\frac{\frac{1}{60}}{{v}^{4}}}}{v}} \]
13. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \color{blue}{\frac{\frac{1}{60}}{{v}^{4}}}}{v}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60}}{{v}^{\color{blue}{\left(2 \cdot 2\right)}}}}{v}} \]

    3. pow-sqr N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60}}{\color{blue}{{v}^{2} \cdot {v}^{2}}}}{v}} \]

    4. lower-*.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60}}{\color{blue}{{v}^{2} \cdot {v}^{2}}}}{v}} \]

    5. unpow2 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60}}{\color{blue}{\left(v \cdot v\right)} \cdot {v}^{2}}}{v}} \]

    6. lower-*.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60}}{\color{blue}{\left(v \cdot v\right)} \cdot {v}^{2}}}{v}} \]

    7. unpow2 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) + \frac{\frac{1}{60}}{\left(v \cdot v\right) \cdot \color{blue}{\left(v \cdot v\right)}}}{v}} \]

    8. lower-*.f32 70.0

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666}{\left(v \cdot v\right) \cdot \color{blue}{\left(v \cdot v\right)}}}{v}} \]

14. Simplified 70.0%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \color{blue}{\frac{0.016666666666666666}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}}{v}} \]

15. Final simplification 70.0%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) + \frac{0.016666666666666666}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}} \]
16. Add Preprocessing

Alternative 8: 63.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ sinTheta\_O \cdot \mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \frac{-sinTheta\_i}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}, \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) \cdot \left(sinTheta\_O \cdot v\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  sinTheta_O
  (fma
   cosTheta_O
   (* cosTheta_i (/ (- sinTheta_i) (fma 2.0 (* v v) 0.3333333333333333)))
   (/
    (* cosTheta_i cosTheta_O)
    (* (+ 2.0 (/ 0.3333333333333333 (* v v))) (* sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return sinTheta_O * fmaf(cosTheta_O, (cosTheta_i * (-sinTheta_i / fmaf(2.0f, (v * v), 0.3333333333333333f))), ((cosTheta_i * cosTheta_O) / ((2.0f + (0.3333333333333333f / (v * v))) * (sinTheta_O * v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(sinTheta_O * fma(cosTheta_O, Float32(cosTheta_i * Float32(Float32(-sinTheta_i) / fma(Float32(2.0), Float32(v * v), Float32(0.3333333333333333)))), Float32(Float32(cosTheta_i * cosTheta_O) / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) * Float32(sinTheta_O * v)))))
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}} \]

   6. lower-*.f32 64.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 64.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 1\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   2. neg-mul-1 N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)\right) + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\left(\color{blue}{sinTheta\_O \cdot \left(\mathsf{neg}\left(\frac{sinTheta\_i}{v}\right)\right)} + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   5. mul-1-neg N/A

      \[\leadsto \frac{\left(sinTheta\_O \cdot \color{blue}{\left(-1 \cdot \frac{sinTheta\_i}{v}\right)} + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, -1 \cdot \frac{sinTheta\_i}{v}, 1\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\mathsf{neg}\left(\frac{sinTheta\_i}{v}\right)}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_i}{\mathsf{neg}\left(v\right)}}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_i}{\mathsf{neg}\left(v\right)}}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   10. lower-neg.f32 64.0

       \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{\color{blue}{-v}}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

8. Simplified 64.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{-v}, 1\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

9. Taylor expanded in sinTheta_O around inf

   \[\leadsto \color{blue}{sinTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{sinTheta\_O \cdot \left(v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right)} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \color{blue}{sinTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{sinTheta\_O \cdot \left(v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right)} \]

    2. mul-1-neg N/A

       \[\leadsto sinTheta\_O \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{sinTheta\_O \cdot \left(v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]

    3. associate-/l* N/A

       \[\leadsto sinTheta\_O \cdot \left(\left(\mathsf{neg}\left(\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i \cdot sinTheta\_i}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{sinTheta\_O \cdot \left(v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]

    4. distribute-rgt-neg-in N/A

       \[\leadsto sinTheta\_O \cdot \left(\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot sinTheta\_i}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{sinTheta\_O \cdot \left(v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]

    5. lower-fma.f32 N/A

       \[\leadsto sinTheta\_O \cdot \color{blue}{\mathsf{fma}\left(cosTheta\_O, \mathsf{neg}\left(\frac{cosTheta\_i \cdot sinTheta\_i}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right), \frac{cosTheta\_O \cdot cosTheta\_i}{sinTheta\_O \cdot \left(v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)}\right)} \]

11. Simplified 64.0%

    \[\leadsto \color{blue}{sinTheta\_O \cdot \mathsf{fma}\left(cosTheta\_O, -cosTheta\_i \cdot \frac{sinTheta\_i}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}, \frac{cosTheta\_O \cdot cosTheta\_i}{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) \cdot \left(v \cdot sinTheta\_O\right)}\right)} \]

12. Final simplification 64.0%

    \[\leadsto sinTheta\_O \cdot \mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \frac{-sinTheta\_i}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}, \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right) \cdot \left(sinTheta\_O \cdot v\right)}\right) \]
13. Add Preprocessing

Alternative 9: 70.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{2 + \frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v}}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_i cosTheta_O) v)
  (*
   v
   (/
    (+ 2.0 (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v)))
    v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) / (v * ((2.0f + ((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v))) / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = ((costheta_i * costheta_o) / v) / (v * ((2.0e0 + ((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v))) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) / Float32(v * Float32(Float32(Float32(2.0) + Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v))) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) / (v * ((single(2.0) + ((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v))) / v));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. rec-exp N/A

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

   6. lower-exp.f32 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f32 98.5

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

5. Simplified 98.5%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Simplified 98.5%

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

9. Taylor expanded in v around -inf

   \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)}{v}}} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}}{v}} \]

    3. sub-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)}\right)}{v}} \]

    4. distribute-neg-in N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}}{v}} \]

    5. mul-1-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}{v}} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}{v}} \]

    7. sub-neg N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{\color{blue}{\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - \left(\mathsf{neg}\left(2\right)\right)}}{v}} \]

    8. lower-/.f32 N/A

       \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - \left(\mathsf{neg}\left(2\right)\right)}{v}}} \]

11. Simplified 70.0%

    \[\leadsto \frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \color{blue}{\frac{2 + \frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v}}{v}}} \]

12. Final simplification 70.0%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \frac{2 + \frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v}}{v}} \]
13. Add Preprocessing

Alternative 10: 63.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \left(-sinTheta\_i\right), \frac{cosTheta\_i}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}, \frac{cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (fma
   (* sinTheta_O (- sinTheta_i))
   (/ cosTheta_i (fma 2.0 (* v v) 0.3333333333333333))
   (/ cosTheta_i (* v (+ 2.0 (/ 0.3333333333333333 (* v v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * fmaf((sinTheta_O * -sinTheta_i), (cosTheta_i / fmaf(2.0f, (v * v), 0.3333333333333333f)), (cosTheta_i / (v * (2.0f + (0.3333333333333333f / (v * v))))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * fma(Float32(sinTheta_O * Float32(-sinTheta_i)), Float32(cosTheta_i / fma(Float32(2.0), Float32(v * v), Float32(0.3333333333333333))), Float32(cosTheta_i / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))))
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}} \]

   6. lower-*.f32 64.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 64.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

   2. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

   4. associate-/l* N/A

      \[\leadsto cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) + \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) + \frac{cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{{v}^{2} \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) + \frac{cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)} \]

8. Simplified 64.0%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \mathsf{fma}\left(-sinTheta\_O \cdot sinTheta\_i, \frac{cosTheta\_i}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}, \frac{cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}\right)} \]

9. Final simplification 64.0%

   \[\leadsto cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \left(-sinTheta\_i\right), \frac{cosTheta\_i}{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}, \frac{cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}\right) \]
10. Add Preprocessing

Alternative 11: 63.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, -cosTheta\_i, cosTheta\_i\right)}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   cosTheta_O
   (fma (* sinTheta_O (/ sinTheta_i v)) (- cosTheta_i) cosTheta_i))
  (* v (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * fmaf((sinTheta_O * (sinTheta_i / v)), -cosTheta_i, cosTheta_i)) / (v * (2.0f + (0.3333333333333333f / (v * v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * fma(Float32(sinTheta_O * Float32(sinTheta_i / v)), Float32(-cosTheta_i), cosTheta_i)) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}} \]

   6. lower-*.f32 64.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 64.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\color{blue}{\left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 1\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   2. neg-mul-1 N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)} + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\left(\left(\mathsf{neg}\left(\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)\right) + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\left(\color{blue}{sinTheta\_O \cdot \left(\mathsf{neg}\left(\frac{sinTheta\_i}{v}\right)\right)} + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   5. mul-1-neg N/A

      \[\leadsto \frac{\left(sinTheta\_O \cdot \color{blue}{\left(-1 \cdot \frac{sinTheta\_i}{v}\right)} + 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, -1 \cdot \frac{sinTheta\_i}{v}, 1\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\mathsf{neg}\left(\frac{sinTheta\_i}{v}\right)}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_i}{\mathsf{neg}\left(v\right)}}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_i}{\mathsf{neg}\left(v\right)}}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]

   10. lower-neg.f32 64.0

       \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{\color{blue}{-v}}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

8. Simplified 64.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{-v}, 1\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

9. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
10. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

    2. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    3. +-commutative N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + 1\right)}\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    4. distribute-rgt-in N/A

       \[\leadsto \frac{cosTheta\_O \cdot \color{blue}{\left(\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \cdot cosTheta\_i + 1 \cdot cosTheta\_i\right)}}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(\color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot -1\right)} \cdot cosTheta\_i + 1 \cdot cosTheta\_i\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    6. associate-*l* N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \left(-1 \cdot cosTheta\_i\right)} + 1 \cdot cosTheta\_i\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    7. *-lft-identity N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \left(-1 \cdot cosTheta\_i\right) + \color{blue}{cosTheta\_i}\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    8. lower-fma.f32 N/A

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

    9. associate-/l* N/A

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

    10. lower-*.f32 N/A

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

    11. lower-/.f32 N/A

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

    12. neg-mul-1 N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \color{blue}{\mathsf{neg}\left(cosTheta\_i\right)}, cosTheta\_i\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    13. lower-neg.f32 N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \color{blue}{\mathsf{neg}\left(cosTheta\_i\right)}, cosTheta\_i\right)}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

    14. lower-*.f32 N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_i\right)}{\color{blue}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

    15. lower-+.f32 N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_i\right)}{v \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

    16. associate-*r/ N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_i\right)}{v \cdot \left(2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}\right)} \]

    17. metadata-eval N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_i\right)}{v \cdot \left(2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}\right)} \]

    18. lower-/.f32 N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_i\right)}{v \cdot \left(2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}\right)} \]

    19. unpow2 N/A

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_i\right)}{v \cdot \left(2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}\right)} \]

    20. lower-*.f32 64.0

        \[\leadsto \frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, -cosTheta\_i, cosTheta\_i\right)}{v \cdot \left(2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}\right)} \]

11. Simplified 64.0%

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \mathsf{fma}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}, -cosTheta\_i, cosTheta\_i\right)}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}} \]
12. Add Preprocessing

Alternative 12: 63.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i cosTheta_O) (* v (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) / (v * (2.0f + (0.3333333333333333f / (v * v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = (costheta_i * costheta_o) / (v * (2.0e0 + (0.3333333333333333e0 / (v * v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) / (v * (single(2.0) + (single(0.3333333333333333) / (v * v))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}} \]

   6. lower-*.f32 64.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 64.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

6. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

   4. lower-+.f32 N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}\right)} \]

   9. lower-*.f32 64.0

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}\right)} \]

8. Simplified 64.0%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}} \]

9. Final simplification 64.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \]
10. Add Preprocessing

Alternative 13: 58.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = (costheta_i * (costheta_o * 0.5e0)) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Simplified 58.3%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\color{blue}{1} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2} \]
7. Step-by-step derivation

8. Simplified 58.3%

   \[\leadsto \frac{\color{blue}{1} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2} \]

9. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

    2. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i}}{v} \]

    4. *-commutative N/A

       \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)}}{v} \]

    5. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)}}{v} \]

    6. *-commutative N/A

       \[\leadsto \frac{cosTheta\_i \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{1}{2}\right)}}{v} \]

    7. lower-*.f32 58.4

       \[\leadsto \frac{cosTheta\_i \cdot \color{blue}{\left(cosTheta\_O \cdot 0.5\right)}}{v} \]

11. Simplified 58.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
12. Add Preprocessing

Alternative 14: 58.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}{v} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = ((costheta_i * costheta_o) * 0.5e0) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Simplified 58.3%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}}{v} \]

   4. lower-*.f32 58.4

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}}{v} \]

8. Simplified 58.4%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

9. Final simplification 58.4%

   \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}{v} \]
10. Add Preprocessing

Alternative 15: 56.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ v \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* v (* cosTheta_O (* cosTheta_i 3.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return v * (cosTheta_O * (cosTheta_i * 3.0f));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    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 = v * (costheta_o * (costheta_i * 3.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(v * Float32(cosTheta_O * Float32(cosTheta_i * Float32(3.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = v * (cosTheta_O * (cosTheta_i * single(3.0)));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}} \]

   6. lower-*.f32 64.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

5. Simplified 64.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\color{blue}{1} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{v \cdot v}} \]
7. Step-by-step derivation

8. Simplified 64.0%

   \[\leadsto \frac{\color{blue}{1} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

9. Taylor expanded in v around 0

   \[\leadsto \color{blue}{3 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto 3 \cdot \color{blue}{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot v\right)} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right) \cdot v} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{v \cdot \left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

    4. lower-*.f32 N/A

       \[\leadsto \color{blue}{v \cdot \left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

    5. *-commutative N/A

       \[\leadsto v \cdot \color{blue}{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 3\right)} \]

    6. associate-*l* N/A

       \[\leadsto v \cdot \color{blue}{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot 3\right)\right)} \]

    7. *-commutative N/A

       \[\leadsto v \cdot \left(cosTheta\_O \cdot \color{blue}{\left(3 \cdot cosTheta\_i\right)}\right) \]

    8. lower-*.f32 N/A

       \[\leadsto v \cdot \color{blue}{\left(cosTheta\_O \cdot \left(3 \cdot cosTheta\_i\right)\right)} \]

    9. *-commutative N/A

       \[\leadsto v \cdot \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot 3\right)}\right) \]

    10. lower-*.f32 56.7

        \[\leadsto v \cdot \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot 3\right)}\right) \]

11. Simplified 56.7%

    \[\leadsto \color{blue}{v \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot 3\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))