HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.4%

Time: 17.7s

Alternatives: 18

Speedup: N/A×

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: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return fabsf((0.5f / sinhf((-1.0f / v)))) * ((cosTheta_i * (cosTheta_O / v)) * (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 = abs((0.5e0 / sinh(((-1.0e0) / v)))) * ((costheta_i * (costheta_o / v)) * (1.0e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

8. Simplified 98.6%

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

   1. add-sqr-sqrt 98.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   2. sqrt-unprod 98.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   3. inv-pow 98.6%

      \[\leadsto \sqrt{\color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   4. inv-pow 98.6%

      \[\leadsto \sqrt{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1} \cdot \color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   5. pow-prod-up 98.6%

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

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-2}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
11. Step-by-step derivation

    1. metadata-eval 98.6%

       \[\leadsto \sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    2. pow-sqr 98.6%

       \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    3. unpow-1 98.6%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    4. unpow-1 98.6%

       \[\leadsto \sqrt{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)} \cdot \color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    5. rem-sqrt-square 98.6%

       \[\leadsto \color{blue}{\left|\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    6. associate-/r* 98.6%

       \[\leadsto \left|\color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{-1}{v}\right)}}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    7. metadata-eval 98.6%

       \[\leadsto \left|\frac{\color{blue}{0.5}}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

12. Simplified 98.6%

    \[\leadsto \color{blue}{\left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

13. Final simplification 98.6%

    \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

Alternative 2: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \cdot \left(\frac{0.009722222222222222}{{v}^{3}} + \left(0.5 \cdot v - \frac{0.08333333333333333}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.5)
   (* (/ cosTheta_O (pow v 2.0)) (/ cosTheta_i (+ -1.0 (exp (/ 1.0 v)))))
   (*
    (* (* cosTheta_i (/ cosTheta_O v)) (/ 1.0 v))
    (+
     (/ 0.009722222222222222 (pow v 3.0))
     (- (* 0.5 v) (/ 0.08333333333333333 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.5f) {
		tmp = (cosTheta_O / powf(v, 2.0f)) * (cosTheta_i / (-1.0f + expf((1.0f / v))));
	} else {
		tmp = ((cosTheta_i * (cosTheta_O / v)) * (1.0f / v)) * ((0.009722222222222222f / powf(v, 3.0f)) + ((0.5f * v) - (0.08333333333333333f / 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 <= 0.5e0) then
        tmp = (costheta_o / (v ** 2.0e0)) * (costheta_i / ((-1.0e0) + exp((1.0e0 / v))))
    else
        tmp = ((costheta_i * (costheta_o / v)) * (1.0e0 / v)) * ((0.009722222222222222e0 / (v ** 3.0e0)) + ((0.5e0 * v) - (0.08333333333333333e0 / 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(0.5))
		tmp = Float32(Float32(cosTheta_O / (v ^ Float32(2.0))) * Float32(cosTheta_i / Float32(Float32(-1.0) + exp(Float32(Float32(1.0) / v)))));
	else
		tmp = Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(1.0) / v)) * Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(3.0))) + Float32(Float32(Float32(0.5) * v) - Float32(Float32(0.08333333333333333) / 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(0.5))
		tmp = (cosTheta_O / (v ^ single(2.0))) * (cosTheta_i / (single(-1.0) + exp((single(1.0) / v))));
	else
		tmp = ((cosTheta_i * (cosTheta_O / v)) * (single(1.0) / v)) * ((single(0.009722222222222222) / (v ^ single(3.0))) + ((single(0.5) * v) - (single(0.08333333333333333) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 98.3%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      2. times-frac 98.0%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

      3. distribute-neg-frac 98.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

      4. distribute-rgt-neg-out 98.0%

         \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

      5. associate-*l/ 98.0%

         \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

      6. *-commutative 98.0%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

      7. associate-*l/ 97.9%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot cosTheta_i}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   3. Simplified 97.9%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   4. Taylor expanded in sinTheta_i around 0 98.2%

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

      1. times-frac 98.3%

         \[\leadsto \color{blue}{\frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

      2. rec-exp 98.3%

         \[\leadsto \frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}} \]

      3. distribute-neg-frac 98.3%

         \[\leadsto \frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

      4. metadata-eval 98.3%

         \[\leadsto \frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

   6. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

   7. Taylor expanded in v around inf 72.0%

      \[\leadsto \frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{e^{\frac{1}{v}} - \color{blue}{1}} \]

   if 0.5 < v

   1. Initial program 99.0%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. times-frac 98.9%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

      2. exp-neg 98.9%

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      3. *-commutative 98.9%

         \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      4. exp-neg 98.9%

         \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      5. distribute-neg-frac 98.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      6. *-commutative 98.9%

         \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      7. distribute-rgt-neg-out 98.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      8. associate-/l* 98.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      9. associate-/l* 98.8%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
   4. Step-by-step derivation

      1. div-inv 99.0%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

      2. div-inv 98.8%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

      3. clear-num 98.9%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

   6. Taylor expanded in sinTheta_i around 0 99.0%

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

      1. rec-exp 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. metadata-eval 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in v around inf 73.8%

      \[\leadsto \color{blue}{\left(\left(0.009722222222222222 \cdot \frac{1}{{v}^{3}} + 0.5 \cdot v\right) - 0.08333333333333333 \cdot \frac{1}{v}\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 73.8%

          \[\leadsto \color{blue}{\left(0.009722222222222222 \cdot \frac{1}{{v}^{3}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       2. associate-*r/ 73.8%

          \[\leadsto \left(\color{blue}{\frac{0.009722222222222222 \cdot 1}{{v}^{3}}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       3. metadata-eval 73.8%

          \[\leadsto \left(\frac{\color{blue}{0.009722222222222222}}{{v}^{3}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       4. *-commutative 73.8%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(\color{blue}{v \cdot 0.5} - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       5. associate-*r/ 73.8%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \color{blue}{\frac{0.08333333333333333 \cdot 1}{v}}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       6. metadata-eval 73.8%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{\color{blue}{0.08333333333333333}}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   11. Simplified 73.8%

       \[\leadsto \color{blue}{\left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{0.08333333333333333}{v}\right)\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\frac{cosTheta_O}{{v}^{2}} \cdot \frac{cosTheta_i}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \cdot \left(\frac{0.009722222222222222}{{v}^{3}} + \left(0.5 \cdot v - \frac{0.08333333333333333}{v}\right)\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \end{array} \]

FPCore

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

C

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

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 = abs((0.5e0 / sinh(((-1.0e0) / v)))) * (costheta_i / (v * (v / costheta_o)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

8. Simplified 98.6%

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

   1. add-sqr-sqrt 98.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   2. sqrt-unprod 98.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   3. inv-pow 98.6%

      \[\leadsto \sqrt{\color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   4. inv-pow 98.6%

      \[\leadsto \sqrt{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1} \cdot \color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   5. pow-prod-up 98.6%

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

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-2}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
11. Step-by-step derivation

    1. metadata-eval 98.6%

       \[\leadsto \sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    2. pow-sqr 98.6%

       \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    3. unpow-1 98.6%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    4. unpow-1 98.6%

       \[\leadsto \sqrt{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)} \cdot \color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    5. rem-sqrt-square 98.6%

       \[\leadsto \color{blue}{\left|\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    6. associate-/r* 98.6%

       \[\leadsto \left|\color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{-1}{v}\right)}}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    7. metadata-eval 98.6%

       \[\leadsto \left|\frac{\color{blue}{0.5}}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

12. Simplified 98.6%

    \[\leadsto \color{blue}{\left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
13. Step-by-step derivation

    1. associate-*r/ 98.7%

       \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \cdot \frac{1}{v}\right) \]

    2. associate-/l* 98.7%

       \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \cdot \frac{1}{v}\right) \]

    3. div-inv 98.4%

       \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \color{blue}{\frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]

    4. associate-/l/ 98.4%

       \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

14. Applied egg-rr 98.4%

    \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

15. Final simplification 98.4%

    \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return fabsf((0.5f / sinhf((-1.0f / v)))) * ((cosTheta_i * (cosTheta_O / 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 = abs((0.5e0 / sinh(((-1.0e0) / v)))) * ((costheta_i * (costheta_o / v)) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

8. Simplified 98.6%

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

   1. add-sqr-sqrt 98.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   2. sqrt-unprod 98.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   3. inv-pow 98.6%

      \[\leadsto \sqrt{\color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   4. inv-pow 98.6%

      \[\leadsto \sqrt{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1} \cdot \color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   5. pow-prod-up 98.6%

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

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-2}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
11. Step-by-step derivation

    1. metadata-eval 98.6%

       \[\leadsto \sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    2. pow-sqr 98.6%

       \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    3. unpow-1 98.6%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    4. unpow-1 98.6%

       \[\leadsto \sqrt{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)} \cdot \color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    5. rem-sqrt-square 98.6%

       \[\leadsto \color{blue}{\left|\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    6. associate-/r* 98.6%

       \[\leadsto \left|\color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{-1}{v}\right)}}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    7. metadata-eval 98.6%

       \[\leadsto \left|\frac{\color{blue}{0.5}}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

12. Simplified 98.6%

    \[\leadsto \color{blue}{\left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
13. Step-by-step derivation

    1. un-div-inv 98.4%

       \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}} \]

14. Applied egg-rr 98.4%

    \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}} \]

15. Final simplification 98.4%

    \[\leadsto \left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v} \]

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \frac{cosTheta_i}{\frac{v}{\frac{cosTheta_O}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / fabsf(sinhf((-1.0f / v)))) * (cosTheta_i / (v / (cosTheta_O / 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 = (0.5e0 / abs(sinh(((-1.0e0) / v)))) * (costheta_i / (v / (costheta_o / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

8. Simplified 98.6%

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

   1. add-sqr-sqrt 98.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   2. sqrt-unprod 98.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   3. inv-pow 98.6%

      \[\leadsto \sqrt{\color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}} \cdot \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   4. inv-pow 98.6%

      \[\leadsto \sqrt{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1} \cdot \color{blue}{{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   5. pow-prod-up 98.6%

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

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-2}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
11. Step-by-step derivation

    1. metadata-eval 98.6%

       \[\leadsto \sqrt{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    2. pow-sqr 98.6%

       \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    3. unpow-1 98.6%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}} \cdot {\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}^{-1}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    4. unpow-1 98.6%

       \[\leadsto \sqrt{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)} \cdot \color{blue}{\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    5. rem-sqrt-square 98.6%

       \[\leadsto \color{blue}{\left|\frac{1}{2 \cdot \sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    6. associate-/r* 98.6%

       \[\leadsto \left|\color{blue}{\frac{\frac{1}{2}}{\sinh \left(\frac{-1}{v}\right)}}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

    7. metadata-eval 98.6%

       \[\leadsto \left|\frac{\color{blue}{0.5}}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

12. Simplified 98.6%

    \[\leadsto \color{blue}{\left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
13. Step-by-step derivation

    1. expm1-log1p-u 98.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)\right)\right)} \]

    2. expm1-udef 52.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|\frac{0.5}{\sinh \left(\frac{-1}{v}\right)}\right| \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)\right)} - 1} \]

    3. fabs-div 52.5%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|0.5\right|}{\left|\sinh \left(\frac{-1}{v}\right)\right|}} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)\right)} - 1 \]

    4. metadata-eval 52.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)\right)} - 1 \]

    5. un-div-inv 52.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}}\right)} - 1 \]

14. Applied egg-rr 52.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 98.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}\right)\right)} \]

    2. expm1-log1p 98.4%

       \[\leadsto \color{blue}{\frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}} \]

    3. associate-/l* 98.5%

       \[\leadsto \frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{\frac{cosTheta_O}{v}}}} \]

16. Simplified 98.5%

    \[\leadsto \color{blue}{\frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \frac{cosTheta_i}{\frac{v}{\frac{cosTheta_O}{v}}}} \]

17. Final simplification 98.5%

    \[\leadsto \frac{0.5}{\left|\sinh \left(\frac{-1}{v}\right)\right|} \cdot \frac{cosTheta_i}{\frac{v}{\frac{cosTheta_O}{v}}} \]

Alternative 6: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\\ \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;t_0 \cdot \frac{1}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{0.009722222222222222}{{v}^{3}} + \left(0.5 \cdot v - \frac{0.08333333333333333}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* (* cosTheta_i (/ cosTheta_O v)) (/ 1.0 v))))
   (if (<= v 0.5)
     (* t_0 (/ 1.0 (+ -1.0 (exp (/ 1.0 v)))))
     (*
      t_0
      (+
       (/ 0.009722222222222222 (pow v 3.0))
       (- (* 0.5 v) (/ 0.08333333333333333 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)) * (1.0f / v);
	float tmp;
	if (v <= 0.5f) {
		tmp = t_0 * (1.0f / (-1.0f + expf((1.0f / v))));
	} else {
		tmp = t_0 * ((0.009722222222222222f / powf(v, 3.0f)) + ((0.5f * v) - (0.08333333333333333f / 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)) * (1.0e0 / v)
    if (v <= 0.5e0) then
        tmp = t_0 * (1.0e0 / ((-1.0e0) + exp((1.0e0 / v))))
    else
        tmp = t_0 * ((0.009722222222222222e0 / (v ** 3.0e0)) + ((0.5e0 * v) - (0.08333333333333333e0 / 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 * Float32(cosTheta_O / v)) * Float32(Float32(1.0) / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.5))
		tmp = Float32(t_0 * Float32(Float32(1.0) / Float32(Float32(-1.0) + exp(Float32(Float32(1.0) / v)))));
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(3.0))) + Float32(Float32(Float32(0.5) * v) - Float32(Float32(0.08333333333333333) / 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)) * (single(1.0) / v);
	tmp = single(0.0);
	if (v <= single(0.5))
		tmp = t_0 * (single(1.0) / (single(-1.0) + exp((single(1.0) / v))));
	else
		tmp = t_0 * ((single(0.009722222222222222) / (v ^ single(3.0))) + ((single(0.5) * v) - (single(0.08333333333333333) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.5

   1. Initial program 98.3%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. times-frac 98.1%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

      2. exp-neg 98.1%

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      3. *-commutative 98.1%

         \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      4. exp-neg 98.1%

         \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      5. distribute-neg-frac 98.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      6. *-commutative 98.1%

         \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      7. distribute-rgt-neg-out 98.1%

         \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      8. associate-/l* 98.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      9. associate-/l* 98.2%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
   4. Step-by-step derivation

      1. div-inv 98.5%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

      2. div-inv 98.5%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

      3. clear-num 98.5%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

   5. Applied egg-rr 98.5%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

   6. Taylor expanded in sinTheta_i around 0 98.4%

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

      1. rec-exp 98.4%

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

      2. distribute-neg-frac 98.4%

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

      3. metadata-eval 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in v around inf 72.0%

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

   if 0.5 < v

   1. Initial program 99.0%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. times-frac 98.9%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

      2. exp-neg 98.9%

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      3. *-commutative 98.9%

         \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      4. exp-neg 98.9%

         \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      5. distribute-neg-frac 98.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      6. *-commutative 98.9%

         \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      7. distribute-rgt-neg-out 98.9%

         \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      8. associate-/l* 98.9%

         \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      9. associate-/l* 98.8%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
   4. Step-by-step derivation

      1. div-inv 99.0%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

      2. div-inv 98.8%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

      3. clear-num 98.9%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

   6. Taylor expanded in sinTheta_i around 0 99.0%

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

      1. rec-exp 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. metadata-eval 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in v around inf 73.8%

      \[\leadsto \color{blue}{\left(\left(0.009722222222222222 \cdot \frac{1}{{v}^{3}} + 0.5 \cdot v\right) - 0.08333333333333333 \cdot \frac{1}{v}\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 73.8%

          \[\leadsto \color{blue}{\left(0.009722222222222222 \cdot \frac{1}{{v}^{3}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       2. associate-*r/ 73.8%

          \[\leadsto \left(\color{blue}{\frac{0.009722222222222222 \cdot 1}{{v}^{3}}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       3. metadata-eval 73.8%

          \[\leadsto \left(\frac{\color{blue}{0.009722222222222222}}{{v}^{3}} + \left(0.5 \cdot v - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       4. *-commutative 73.8%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(\color{blue}{v \cdot 0.5} - 0.08333333333333333 \cdot \frac{1}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       5. associate-*r/ 73.8%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \color{blue}{\frac{0.08333333333333333 \cdot 1}{v}}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

       6. metadata-eval 73.8%

          \[\leadsto \left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{\color{blue}{0.08333333333333333}}{v}\right)\right) \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]

   11. Simplified 73.8%

       \[\leadsto \color{blue}{\left(\frac{0.009722222222222222}{{v}^{3}} + \left(v \cdot 0.5 - \frac{0.08333333333333333}{v}\right)\right)} \cdot \left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \cdot \frac{1}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \cdot \left(\frac{0.009722222222222222}{{v}^{3}} + \left(0.5 \cdot v - \frac{0.08333333333333333}{v}\right)\right)\\ \end{array} \]

Alternative 7: 71.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \cdot \frac{1}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O}{2 + \frac{0.3333333333333333}{{v}^{2}}} \cdot \frac{cosTheta_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   (*
    (* (* cosTheta_i (/ cosTheta_O v)) (/ 1.0 v))
    (/ 1.0 (+ -1.0 (exp (/ 1.0 v)))))
   (*
    (/ cosTheta_O (+ 2.0 (/ 0.3333333333333333 (pow v 2.0))))
    (/ cosTheta_i v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = ((cosTheta_i * (cosTheta_O / v)) * (1.0f / v)) * (1.0f / (-1.0f + expf((1.0f / v))));
	} else {
		tmp = (cosTheta_O / (2.0f + (0.3333333333333333f / powf(v, 2.0f)))) * (cosTheta_i / 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 <= 0.44999998807907104e0) then
        tmp = ((costheta_i * (costheta_o / v)) * (1.0e0 / v)) * (1.0e0 / ((-1.0e0) + exp((1.0e0 / v))))
    else
        tmp = (costheta_o / (2.0e0 + (0.3333333333333333e0 / (v ** 2.0e0)))) * (costheta_i / v)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(1.0) / v)) * Float32(Float32(1.0) / Float32(Float32(-1.0) + exp(Float32(Float32(1.0) / v)))));
	else
		tmp = Float32(Float32(cosTheta_O / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / (v ^ Float32(2.0))))) * Float32(cosTheta_i / 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(0.44999998807907104))
		tmp = ((cosTheta_i * (cosTheta_O / v)) * (single(1.0) / v)) * (single(1.0) / (single(-1.0) + exp((single(1.0) / v))));
	else
		tmp = (cosTheta_O / (single(2.0) + (single(0.3333333333333333) / (v ^ single(2.0))))) * (cosTheta_i / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   1. Initial program 98.2%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. times-frac 98.1%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

      2. exp-neg 98.1%

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      3. *-commutative 98.1%

         \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      4. exp-neg 98.1%

         \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      5. distribute-neg-frac 98.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      6. *-commutative 98.1%

         \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      7. distribute-rgt-neg-out 98.1%

         \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      8. associate-/l* 98.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

      9. associate-/l* 98.1%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
   4. Step-by-step derivation

      1. div-inv 98.4%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

      2. div-inv 98.4%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

      3. clear-num 98.4%

         \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

   6. Taylor expanded in sinTheta_i around 0 98.4%

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

      1. rec-exp 98.4%

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

      2. distribute-neg-frac 98.4%

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

      3. metadata-eval 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in v around inf 72.8%

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

   if 0.449999988 < v

   1. Initial program 99.0%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. Taylor expanded in v around inf 70.2%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.2%

         \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

      2. metadata-eval 70.2%

         \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

   4. Simplified 70.2%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

   5. Taylor expanded in sinTheta_i around 0 70.2%

      \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. metadata-eval 70.2%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot -1}}{{v}^{2}}\right)} \]

      2. unpow2 70.2%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{-1 \cdot -1}{\color{blue}{v \cdot v}}\right)} \]

      3. frac-times 70.2%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\frac{-1}{v} \cdot \frac{-1}{v}\right)}\right)} \]

   7. Applied egg-rr 70.2%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\frac{-1}{v} \cdot \frac{-1}{v}\right)}\right)} \]

   8. Taylor expanded in cosTheta_O around 0 70.2%

      \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right) \cdot v}} \]

      2. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{cosTheta_O}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}} \cdot \frac{cosTheta_i}{v}} \]

      3. associate-*r/ 70.2%

         \[\leadsto \frac{cosTheta_O}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \cdot \frac{cosTheta_i}{v} \]

      4. metadata-eval 70.2%

         \[\leadsto \frac{cosTheta_O}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \cdot \frac{cosTheta_i}{v} \]

   10. Simplified 70.2%

       \[\leadsto \color{blue}{\frac{cosTheta_O}{2 + \frac{0.3333333333333333}{{v}^{2}}} \cdot \frac{cosTheta_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right) \cdot \frac{1}{-1 + e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O}{2 + \frac{0.3333333333333333}{{v}^{2}}} \cdot \frac{cosTheta_i}{v}\\ \end{array} \]

Alternative 8: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + \frac{0.3333333333333333}{{v}^{2}}} \end{array} \]

FPCore

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

C

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

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 = (costheta_o / v) * (costheta_i / (2.0e0 + (0.3333333333333333e0 / (v ** 2.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

2. Taylor expanded in v around inf 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   2. metadata-eval 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
6. Step-by-step derivation

   1. times-frac 63.6%

      \[\leadsto \color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]

   2. associate-*r/ 63.6%

      \[\leadsto \frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   3. metadata-eval 63.6%

      \[\leadsto \frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

7. Simplified 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

8. Final simplification 63.6%

   \[\leadsto \frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{2 + \frac{0.3333333333333333}{{v}^{2}}} \]

Alternative 9: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{{v}^{2}}} \cdot \frac{cosTheta_i}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / (2.0f + (0.3333333333333333f / powf(v, 2.0f)))) * (cosTheta_i / 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_o / (2.0e0 + (0.3333333333333333e0 / (v ** 2.0e0)))) * (costheta_i / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

2. Taylor expanded in v around inf 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   2. metadata-eval 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
6. Step-by-step derivation

   1. metadata-eval 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{\color{blue}{-1 \cdot -1}}{{v}^{2}}\right)} \]

   2. unpow2 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{-1 \cdot -1}{\color{blue}{v \cdot v}}\right)} \]

   3. frac-times 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\frac{-1}{v} \cdot \frac{-1}{v}\right)}\right)} \]

7. Applied egg-rr 63.6%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\frac{-1}{v} \cdot \frac{-1}{v}\right)}\right)} \]

8. Taylor expanded in cosTheta_O around 0 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
9. Step-by-step derivation

   1. *-commutative 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right) \cdot v}} \]

   2. times-frac 63.6%

      \[\leadsto \color{blue}{\frac{cosTheta_O}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}} \cdot \frac{cosTheta_i}{v}} \]

   3. associate-*r/ 63.6%

      \[\leadsto \frac{cosTheta_O}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \cdot \frac{cosTheta_i}{v} \]

   4. metadata-eval 63.6%

      \[\leadsto \frac{cosTheta_O}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \cdot \frac{cosTheta_i}{v} \]

10. Simplified 63.6%

    \[\leadsto \color{blue}{\frac{cosTheta_O}{2 + \frac{0.3333333333333333}{{v}^{2}}} \cdot \frac{cosTheta_i}{v}} \]

11. Final simplification 63.6%

    \[\leadsto \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{{v}^{2}}} \cdot \frac{cosTheta_i}{v} \]

Alternative 10: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

2. Taylor expanded in v around inf 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   2. metadata-eval 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]

6. Taylor expanded in v around 0 63.6%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]
7. Step-by-step derivation

   1. *-commutative 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{v \cdot 2} + 0.3333333333333333 \cdot \frac{1}{v}} \]

   2. fma-def 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\mathsf{fma}\left(v, 2, 0.3333333333333333 \cdot \frac{1}{v}\right)}} \]

   3. associate-*r/ 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}\right)} \]

   4. metadata-eval 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \frac{\color{blue}{0.3333333333333333}}{v}\right)} \]

8. Simplified 63.6%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

9. Final simplification 63.6%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]

Alternative 11: 63.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i \cdot cosTheta_O}{v \cdot \left(2 + \frac{\frac{0.3333333333333333}{v}}{v}\right)} \end{array} \]

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(Float32(0.3333333333333333) / 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 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

2. Taylor expanded in v around inf 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   2. metadata-eval 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
6. Step-by-step derivation

   1. un-div-inv 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + \color{blue}{\frac{0.3333333333333333}{{v}^{2}}}\right)} \]

   2. unpow2 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}\right)} \]

   3. associate-/r* 63.6%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + \color{blue}{\frac{\frac{0.3333333333333333}{v}}{v}}\right)} \]

7. Applied egg-rr 63.6%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + \color{blue}{\frac{\frac{0.3333333333333333}{v}}{v}}\right)} \]

8. Final simplification 63.6%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{v \cdot \left(2 + \frac{\frac{0.3333333333333333}{v}}{v}\right)} \]

Alternative 12: 63.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i \cdot cosTheta_O}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i cosTheta_O) (+ (* v 2.0) (* (/ 1.0 v) 0.3333333333333333))))

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) + ((1.0f / v) * 0.3333333333333333f));
}

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) + ((1.0e0 / v) * 0.3333333333333333e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) / Float32(Float32(v * Float32(2.0)) + Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) / ((v * single(2.0)) + ((single(1.0) / v) * single(0.3333333333333333)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

2. Taylor expanded in v around inf 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}}} \]

   2. metadata-eval 63.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

4. Simplified 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}} \]

5. Taylor expanded in sinTheta_i around 0 63.6%

   \[\leadsto \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]

6. Taylor expanded in v around 0 63.6%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

7. Final simplification 63.6%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \]

Alternative 13: 58.6% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \end{array} \]

FPCore

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

C

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

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 = 0.5e0 * (1.0e0 / ((v / costheta_o) / costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   2. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. associate-*l/ 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-*l/ 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot cosTheta_i}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. associate-*r/ 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

6. Simplified 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]
7. Step-by-step derivation

   1. *-commutative 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \]

   2. associate-/r/ 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

   3. clear-num 58.0%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

8. Applied egg-rr 58.0%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

9. Final simplification 58.0%

   \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]

Alternative 14: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O * (cosTheta_i / 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 = 0.5e0 * (costheta_o * (costheta_i / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   2. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. associate-*l/ 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-*l/ 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot cosTheta_i}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. associate-*r/ 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

6. Simplified 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

7. Final simplification 57.7%

   \[\leadsto 0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]

Alternative 15: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_i * cosTheta_O) / 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 = 0.5e0 * ((costheta_i * costheta_o) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   2. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

   3. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   4. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   5. associate-*l/ 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}}{v} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   7. associate-*l/ 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\color{blue}{\frac{cosTheta_O}{v} \cdot cosTheta_i}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}}{v} \cdot \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

4. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

5. Final simplification 57.7%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Alternative 16: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ cosTheta_O \cdot \left(cosTheta_i \cdot \frac{0.5}{v}\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i * (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_o * (costheta_i * (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
7. Step-by-step derivation

   1. associate-/l* 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

8. Simplified 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

9. Taylor expanded in cosTheta_O around 0 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
10. Step-by-step derivation

    1. associate-/l* 57.7%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

    2. associate-*r/ 57.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_O}{\frac{v}{cosTheta_i}}} \]

    3. associate-*l/ 57.7%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta_i}} \cdot cosTheta_O} \]

    4. *-commutative 57.7%

       \[\leadsto \color{blue}{cosTheta_O \cdot \frac{0.5}{\frac{v}{cosTheta_i}}} \]

    5. associate-/r/ 57.7%

       \[\leadsto cosTheta_O \cdot \color{blue}{\left(\frac{0.5}{v} \cdot cosTheta_i\right)} \]

11. Simplified 57.7%

    \[\leadsto \color{blue}{cosTheta_O \cdot \left(\frac{0.5}{v} \cdot cosTheta_i\right)} \]

12. Final simplification 57.7%

    \[\leadsto cosTheta_O \cdot \left(cosTheta_i \cdot \frac{0.5}{v}\right) \]

Alternative 17: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ cosTheta_i \cdot \left(0.5 \cdot \frac{cosTheta_O}{v}\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * (0.5f * (cosTheta_O / 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 * (0.5e0 * (costheta_o / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
7. Step-by-step derivation

   1. associate-/l* 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

8. Simplified 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]
9. Step-by-step derivation

   1. associate-*r/ 57.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_O}{\frac{v}{cosTheta_i}}} \]

   2. frac-2neg 57.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot cosTheta_O}{-\frac{v}{cosTheta_i}}} \]

   3. distribute-neg-frac 57.7%

      \[\leadsto \frac{-0.5 \cdot cosTheta_O}{\color{blue}{\frac{-v}{cosTheta_i}}} \]

10. Applied egg-rr 57.7%

    \[\leadsto \color{blue}{\frac{-0.5 \cdot cosTheta_O}{\frac{-v}{cosTheta_i}}} \]
11. Step-by-step derivation

    1. associate-/r/ 57.7%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot cosTheta_O}{-v} \cdot cosTheta_i} \]

    2. *-commutative 57.7%

       \[\leadsto \color{blue}{cosTheta_i \cdot \frac{-0.5 \cdot cosTheta_O}{-v}} \]

    3. distribute-lft-neg-in 57.7%

       \[\leadsto cosTheta_i \cdot \frac{\color{blue}{\left(-0.5\right) \cdot cosTheta_O}}{-v} \]

    4. metadata-eval 57.7%

       \[\leadsto cosTheta_i \cdot \frac{\color{blue}{-0.5} \cdot cosTheta_O}{-v} \]

    5. neg-mul-1 57.7%

       \[\leadsto cosTheta_i \cdot \frac{-0.5 \cdot cosTheta_O}{\color{blue}{-1 \cdot v}} \]

    6. times-frac 57.7%

       \[\leadsto cosTheta_i \cdot \color{blue}{\left(\frac{-0.5}{-1} \cdot \frac{cosTheta_O}{v}\right)} \]

    7. metadata-eval 57.7%

       \[\leadsto cosTheta_i \cdot \left(\color{blue}{0.5} \cdot \frac{cosTheta_O}{v}\right) \]

12. Simplified 57.7%

    \[\leadsto \color{blue}{cosTheta_i \cdot \left(0.5 \cdot \frac{cosTheta_O}{v}\right)} \]

13. Final simplification 57.7%

    \[\leadsto cosTheta_i \cdot \left(0.5 \cdot \frac{cosTheta_O}{v}\right) \]

Alternative 18: 58.6% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}} \end{array} \]

FPCore

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

C

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

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 = 0.5e0 / ((v / costheta_i) / costheta_o)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.4%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.4%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.4%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.4%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.4%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]
4. Step-by-step derivation

   1. div-inv 98.7%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot \frac{1}{v}\right)} \]

   2. div-inv 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\color{blue}{\left(cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}\right)} \cdot \frac{1}{v}\right) \]

   3. clear-num 98.6%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{v}\right)} \]

6. Taylor expanded in v around inf 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
7. Step-by-step derivation

   1. associate-/l* 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

8. Simplified 57.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]
9. Step-by-step derivation

   1. clear-num 58.0%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}}} \]

   2. un-div-inv 58.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}}} \]

10. Applied egg-rr 58.0%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}}} \]

11. Final simplification 58.0%

    \[\leadsto \frac{0.5}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}} \]

Reproduce

herbie shell --seed 2023308 
(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)))