Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 67.4% → 90.8%

Time: 12.1s

Alternatives: 16

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-260} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (or (<= t_1 -4e-260) (not (<= t_1 0.0)))
     (fma (- y x) (/ (- z t) (- a t)) x)
     (+ y (* x (/ (- z a) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if ((t_1 <= -4e-260) || !(t_1 <= 0.0)) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = y + (x * ((z - a) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if ((t_1 <= -4e-260) || !(t_1 <= 0.0))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = Float64(y + Float64(x * Float64(Float64(z - a) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e-260], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -3.99999999999999985e-260 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 71.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 71.5%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 89.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 89.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   if -3.99999999999999985e-260 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 4.7%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 4.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 4.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 99.4%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in y around 0 99.4%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}{t}} \]

      2. mul-1-neg 99.4%

         \[\leadsto y + \frac{\color{blue}{-\left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}}{t} \]

      3. associate-*r* 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-1 \cdot a\right) \cdot x}\right)}{t} \]

      4. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-a\right)} \cdot x\right)}{t} \]

      5. cancel-sign-sub 99.4%

         \[\leadsto y + \frac{-\color{blue}{\left(-1 \cdot \left(x \cdot z\right) + a \cdot x\right)}}{t} \]

      6. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{\left(-x \cdot z\right)} + a \cdot x\right)}{t} \]

      7. distribute-rgt-neg-in 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{x \cdot \left(-z\right)} + a \cdot x\right)}{t} \]

      8. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot x\right)}{t} \]

      9. *-commutative 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \left(-1 \cdot z\right) + \color{blue}{x \cdot a}\right)}{t} \]

      10. distribute-lft-in 99.4%

          \[\leadsto y + \frac{-\color{blue}{x \cdot \left(-1 \cdot z + a\right)}}{t} \]

      11. *-lft-identity 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{1 \cdot a}\right)}{t} \]

      12. metadata-eval 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{\left(--1\right)} \cdot a\right)}{t} \]

      13. cancel-sign-sub-inv 99.4%

          \[\leadsto y + \frac{-x \cdot \color{blue}{\left(-1 \cdot z - -1 \cdot a\right)}}{t} \]

      14. distribute-neg-frac 99.4%

          \[\leadsto y + \color{blue}{\left(-\frac{x \cdot \left(-1 \cdot z - -1 \cdot a\right)}{t}\right)} \]

      15. associate-/l* 99.5%

          \[\leadsto y + \left(-\color{blue}{x \cdot \frac{-1 \cdot z - -1 \cdot a}{t}}\right) \]

      16. distribute-lft-out-- 99.5%

          \[\leadsto y + \left(-x \cdot \frac{\color{blue}{-1 \cdot \left(z - a\right)}}{t}\right) \]

      17. associate-*r/ 99.5%

          \[\leadsto y + \left(-x \cdot \color{blue}{\left(-1 \cdot \frac{z - a}{t}\right)}\right) \]

   8. Simplified 99.5%

      \[\leadsto y + \color{blue}{x \cdot \frac{z - a}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -4 \cdot 10^{-260} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y \cdot \left(1 - \frac{x}{y}\right)\right) \cdot \frac{z - t}{a - t}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (* y (- 1.0 (/ x y))) (/ (- z t) (- a t)))))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -2e+250)
     t_1
     (if (<= t_2 -4e-260)
       t_2
       (if (<= t_2 0.0)
         (+ y (* x (/ (- z a) t)))
         (if (<= t_2 5e+287) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * (1.0 - (x / y))) * ((z - t) / (a - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -2e+250) {
		tmp = t_1;
	} else if (t_2 <= -4e-260) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y + (x * ((z - a) / t));
	} else if (t_2 <= 5e+287) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + ((y * (1.0d0 - (x / y))) * ((z - t) / (a - t)))
    t_2 = x + (((y - x) * (z - t)) / (a - t))
    if (t_2 <= (-2d+250)) then
        tmp = t_1
    else if (t_2 <= (-4d-260)) then
        tmp = t_2
    else if (t_2 <= 0.0d0) then
        tmp = y + (x * ((z - a) / t))
    else if (t_2 <= 5d+287) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * (1.0 - (x / y))) * ((z - t) / (a - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -2e+250) {
		tmp = t_1;
	} else if (t_2 <= -4e-260) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y + (x * ((z - a) / t));
	} else if (t_2 <= 5e+287) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y * (1.0 - (x / y))) * ((z - t) / (a - t)))
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_2 <= -2e+250:
		tmp = t_1
	elif t_2 <= -4e-260:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = y + (x * ((z - a) / t))
	elif t_2 <= 5e+287:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * Float64(1.0 - Float64(x / y))) * Float64(Float64(z - t) / Float64(a - t))))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -2e+250)
		tmp = t_1;
	elseif (t_2 <= -4e-260)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(x * Float64(Float64(z - a) / t)));
	elseif (t_2 <= 5e+287)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y * (1.0 - (x / y))) * ((z - t) / (a - t)));
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_2 <= -2e+250)
		tmp = t_1;
	elseif (t_2 <= -4e-260)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = y + (x * ((z - a) / t));
	elseif (t_2 <= 5e+287)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+250], t$95$1, If[LessEqual[t$95$2, -4e-260], t$95$2, If[LessEqual[t$95$2, 0.0], N[(y + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+287], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.9999999999999998e250 or 5e287 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 42.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 42.2%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 87.5%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 87.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 81.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 81.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 81.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   8. Step-by-step derivation

      1. fma-undefine 81.5%

         \[\leadsto \color{blue}{\left(y \cdot \left(1 - \frac{x}{y}\right)\right) \cdot \frac{z - t}{a - t} + x} \]

   9. Applied egg-rr 81.5%

      \[\leadsto \color{blue}{\left(y \cdot \left(1 - \frac{x}{y}\right)\right) \cdot \frac{z - t}{a - t} + x} \]

   if -1.9999999999999998e250 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -3.99999999999999985e-260 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 5e287

   1. Initial program 96.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   if -3.99999999999999985e-260 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 4.7%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 4.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 4.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 99.4%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in y around 0 99.4%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}{t}} \]

      2. mul-1-neg 99.4%

         \[\leadsto y + \frac{\color{blue}{-\left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}}{t} \]

      3. associate-*r* 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-1 \cdot a\right) \cdot x}\right)}{t} \]

      4. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-a\right)} \cdot x\right)}{t} \]

      5. cancel-sign-sub 99.4%

         \[\leadsto y + \frac{-\color{blue}{\left(-1 \cdot \left(x \cdot z\right) + a \cdot x\right)}}{t} \]

      6. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{\left(-x \cdot z\right)} + a \cdot x\right)}{t} \]

      7. distribute-rgt-neg-in 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{x \cdot \left(-z\right)} + a \cdot x\right)}{t} \]

      8. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot x\right)}{t} \]

      9. *-commutative 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \left(-1 \cdot z\right) + \color{blue}{x \cdot a}\right)}{t} \]

      10. distribute-lft-in 99.4%

          \[\leadsto y + \frac{-\color{blue}{x \cdot \left(-1 \cdot z + a\right)}}{t} \]

      11. *-lft-identity 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{1 \cdot a}\right)}{t} \]

      12. metadata-eval 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{\left(--1\right)} \cdot a\right)}{t} \]

      13. cancel-sign-sub-inv 99.4%

          \[\leadsto y + \frac{-x \cdot \color{blue}{\left(-1 \cdot z - -1 \cdot a\right)}}{t} \]

      14. distribute-neg-frac 99.4%

          \[\leadsto y + \color{blue}{\left(-\frac{x \cdot \left(-1 \cdot z - -1 \cdot a\right)}{t}\right)} \]

      15. associate-/l* 99.5%

          \[\leadsto y + \left(-\color{blue}{x \cdot \frac{-1 \cdot z - -1 \cdot a}{t}}\right) \]

      16. distribute-lft-out-- 99.5%

          \[\leadsto y + \left(-x \cdot \frac{\color{blue}{-1 \cdot \left(z - a\right)}}{t}\right) \]

      17. associate-*r/ 99.5%

          \[\leadsto y + \left(-x \cdot \color{blue}{\left(-1 \cdot \frac{z - a}{t}\right)}\right) \]

   8. Simplified 99.5%

      \[\leadsto y + \color{blue}{x \cdot \frac{z - a}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2 \cdot 10^{+250}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \frac{x}{y}\right)\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -4 \cdot 10^{-260}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 5 \cdot 10^{+287}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \frac{x}{y}\right)\right) \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ t_2 := \left(y - x\right) \cdot \left(z - t\right)\\ t_3 := x + \frac{t\_2}{a - t}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-260}:\\ \;\;\;\;x + \frac{1}{\frac{a - t}{t\_2}}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t\_3 \leq 10^{+256}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t)))))
        (t_2 (* (- y x) (- z t)))
        (t_3 (+ x (/ t_2 (- a t)))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -4e-260)
       (+ x (/ 1.0 (/ (- a t) t_2)))
       (if (<= t_3 0.0)
         (+ y (* x (/ (- z a) t)))
         (if (<= t_3 1e+256) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double t_2 = (y - x) * (z - t);
	double t_3 = x + (t_2 / (a - t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -4e-260) {
		tmp = x + (1.0 / ((a - t) / t_2));
	} else if (t_3 <= 0.0) {
		tmp = y + (x * ((z - a) / t));
	} else if (t_3 <= 1e+256) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double t_2 = (y - x) * (z - t);
	double t_3 = x + (t_2 / (a - t));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_3 <= -4e-260) {
		tmp = x + (1.0 / ((a - t) / t_2));
	} else if (t_3 <= 0.0) {
		tmp = y + (x * ((z - a) / t));
	} else if (t_3 <= 1e+256) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	t_2 = (y - x) * (z - t)
	t_3 = x + (t_2 / (a - t))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_1
	elif t_3 <= -4e-260:
		tmp = x + (1.0 / ((a - t) / t_2))
	elif t_3 <= 0.0:
		tmp = y + (x * ((z - a) / t))
	elif t_3 <= 1e+256:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	t_2 = Float64(Float64(y - x) * Float64(z - t))
	t_3 = Float64(x + Float64(t_2 / Float64(a - t)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -4e-260)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(a - t) / t_2)));
	elseif (t_3 <= 0.0)
		tmp = Float64(y + Float64(x * Float64(Float64(z - a) / t)));
	elseif (t_3 <= 1e+256)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	t_2 = (y - x) * (z - t);
	t_3 = x + (t_2 / (a - t));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_1;
	elseif (t_3 <= -4e-260)
		tmp = x + (1.0 / ((a - t) / t_2));
	elseif (t_3 <= 0.0)
		tmp = y + (x * ((z - a) / t));
	elseif (t_3 <= 1e+256)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(t$95$2 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -4e-260], N[(x + N[(1.0 / N[(N[(a - t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(y + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+256], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 1e256 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 39.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 38.1%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* 68.4%

         \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   5. Simplified 68.4%

      \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -3.99999999999999985e-260

   1. Initial program 94.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{a - t}{\left(y - x\right) \cdot \left(z - t\right)}}} \]

      2. inv-pow 94.7%

         \[\leadsto x + \color{blue}{{\left(\frac{a - t}{\left(y - x\right) \cdot \left(z - t\right)}\right)}^{-1}} \]

   4. Applied egg-rr 94.7%

      \[\leadsto x + \color{blue}{{\left(\frac{a - t}{\left(y - x\right) \cdot \left(z - t\right)}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. unpow-1 94.7%

         \[\leadsto x + \color{blue}{\frac{1}{\frac{a - t}{\left(y - x\right) \cdot \left(z - t\right)}}} \]

      2. *-commutative 94.7%

         \[\leadsto x + \frac{1}{\frac{a - t}{\color{blue}{\left(z - t\right) \cdot \left(y - x\right)}}} \]

   6. Simplified 94.7%

      \[\leadsto x + \color{blue}{\frac{1}{\frac{a - t}{\left(z - t\right) \cdot \left(y - x\right)}}} \]

   if -3.99999999999999985e-260 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 4.7%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 4.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 4.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 99.4%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in y around 0 99.4%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}{t}} \]

      2. mul-1-neg 99.4%

         \[\leadsto y + \frac{\color{blue}{-\left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}}{t} \]

      3. associate-*r* 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-1 \cdot a\right) \cdot x}\right)}{t} \]

      4. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-a\right)} \cdot x\right)}{t} \]

      5. cancel-sign-sub 99.4%

         \[\leadsto y + \frac{-\color{blue}{\left(-1 \cdot \left(x \cdot z\right) + a \cdot x\right)}}{t} \]

      6. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{\left(-x \cdot z\right)} + a \cdot x\right)}{t} \]

      7. distribute-rgt-neg-in 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{x \cdot \left(-z\right)} + a \cdot x\right)}{t} \]

      8. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot x\right)}{t} \]

      9. *-commutative 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \left(-1 \cdot z\right) + \color{blue}{x \cdot a}\right)}{t} \]

      10. distribute-lft-in 99.4%

          \[\leadsto y + \frac{-\color{blue}{x \cdot \left(-1 \cdot z + a\right)}}{t} \]

      11. *-lft-identity 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{1 \cdot a}\right)}{t} \]

      12. metadata-eval 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{\left(--1\right)} \cdot a\right)}{t} \]

      13. cancel-sign-sub-inv 99.4%

          \[\leadsto y + \frac{-x \cdot \color{blue}{\left(-1 \cdot z - -1 \cdot a\right)}}{t} \]

      14. distribute-neg-frac 99.4%

          \[\leadsto y + \color{blue}{\left(-\frac{x \cdot \left(-1 \cdot z - -1 \cdot a\right)}{t}\right)} \]

      15. associate-/l* 99.5%

          \[\leadsto y + \left(-\color{blue}{x \cdot \frac{-1 \cdot z - -1 \cdot a}{t}}\right) \]

      16. distribute-lft-out-- 99.5%

          \[\leadsto y + \left(-x \cdot \frac{\color{blue}{-1 \cdot \left(z - a\right)}}{t}\right) \]

      17. associate-*r/ 99.5%

          \[\leadsto y + \left(-x \cdot \color{blue}{\left(-1 \cdot \frac{z - a}{t}\right)}\right) \]

   8. Simplified 99.5%

      \[\leadsto y + \color{blue}{x \cdot \frac{z - a}{t}} \]

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 1e256

   1. Initial program 98.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -4 \cdot 10^{-260}:\\ \;\;\;\;x + \frac{1}{\frac{a - t}{\left(y - x\right) \cdot \left(z - t\right)}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 10^{+256}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t\_2 \leq 10^{+256}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t)))))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -4e-260)
       t_2
       (if (<= t_2 0.0)
         (+ y (* x (/ (- z a) t)))
         (if (<= t_2 1e+256) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -4e-260) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y + (x * ((z - a) / t));
	} else if (t_2 <= 1e+256) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -4e-260) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = y + (x * ((z - a) / t));
	} else if (t_2 <= 1e+256) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -4e-260:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = y + (x * ((z - a) / t))
	elif t_2 <= 1e+256:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -4e-260)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(x * Float64(Float64(z - a) / t)));
	elseif (t_2 <= 1e+256)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -4e-260)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = y + (x * ((z - a) / t));
	elseif (t_2 <= 1e+256)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -4e-260], t$95$2, If[LessEqual[t$95$2, 0.0], N[(y + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+256], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 1e256 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 39.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 38.1%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* 68.4%

         \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   5. Simplified 68.4%

      \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -3.99999999999999985e-260 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 1e256

   1. Initial program 96.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   if -3.99999999999999985e-260 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 4.7%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 4.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 4.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 99.4%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in y around 0 99.4%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}{t}} \]

      2. mul-1-neg 99.4%

         \[\leadsto y + \frac{\color{blue}{-\left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}}{t} \]

      3. associate-*r* 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-1 \cdot a\right) \cdot x}\right)}{t} \]

      4. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-a\right)} \cdot x\right)}{t} \]

      5. cancel-sign-sub 99.4%

         \[\leadsto y + \frac{-\color{blue}{\left(-1 \cdot \left(x \cdot z\right) + a \cdot x\right)}}{t} \]

      6. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{\left(-x \cdot z\right)} + a \cdot x\right)}{t} \]

      7. distribute-rgt-neg-in 99.4%

         \[\leadsto y + \frac{-\left(\color{blue}{x \cdot \left(-z\right)} + a \cdot x\right)}{t} \]

      8. mul-1-neg 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot x\right)}{t} \]

      9. *-commutative 99.4%

         \[\leadsto y + \frac{-\left(x \cdot \left(-1 \cdot z\right) + \color{blue}{x \cdot a}\right)}{t} \]

      10. distribute-lft-in 99.4%

          \[\leadsto y + \frac{-\color{blue}{x \cdot \left(-1 \cdot z + a\right)}}{t} \]

      11. *-lft-identity 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{1 \cdot a}\right)}{t} \]

      12. metadata-eval 99.4%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{\left(--1\right)} \cdot a\right)}{t} \]

      13. cancel-sign-sub-inv 99.4%

          \[\leadsto y + \frac{-x \cdot \color{blue}{\left(-1 \cdot z - -1 \cdot a\right)}}{t} \]

      14. distribute-neg-frac 99.4%

          \[\leadsto y + \color{blue}{\left(-\frac{x \cdot \left(-1 \cdot z - -1 \cdot a\right)}{t}\right)} \]

      15. associate-/l* 99.5%

          \[\leadsto y + \left(-\color{blue}{x \cdot \frac{-1 \cdot z - -1 \cdot a}{t}}\right) \]

      16. distribute-lft-out-- 99.5%

          \[\leadsto y + \left(-x \cdot \frac{\color{blue}{-1 \cdot \left(z - a\right)}}{t}\right) \]

      17. associate-*r/ 99.5%

          \[\leadsto y + \left(-x \cdot \color{blue}{\left(-1 \cdot \frac{z - a}{t}\right)}\right) \]

   8. Simplified 99.5%

      \[\leadsto y + \color{blue}{x \cdot \frac{z - a}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-124}:\\ \;\;\;\;x + x \cdot \frac{z - t}{t - a}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-75}:\\ \;\;\;\;y + \left(z - a\right) \cdot \frac{x - y}{t}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (<= y -8.8e-61)
     t_1
     (if (<= y 1.38e-124)
       (+ x (* x (/ (- z t) (- t a))))
       (if (<= y 8.4e-75)
         (+ y (* (- z a) (/ (- x y) t)))
         (if (<= y 2.1e-49) (+ x (* z (/ (- y x) a))) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y <= -8.8e-61) {
		tmp = t_1;
	} else if (y <= 1.38e-124) {
		tmp = x + (x * ((z - t) / (t - a)));
	} else if (y <= 8.4e-75) {
		tmp = y + ((z - a) * ((x - y) / t));
	} else if (y <= 2.1e-49) {
		tmp = x + (z * ((y - x) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y <= (-8.8d-61)) then
        tmp = t_1
    else if (y <= 1.38d-124) then
        tmp = x + (x * ((z - t) / (t - a)))
    else if (y <= 8.4d-75) then
        tmp = y + ((z - a) * ((x - y) / t))
    else if (y <= 2.1d-49) then
        tmp = x + (z * ((y - x) / a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y <= -8.8e-61) {
		tmp = t_1;
	} else if (y <= 1.38e-124) {
		tmp = x + (x * ((z - t) / (t - a)));
	} else if (y <= 8.4e-75) {
		tmp = y + ((z - a) * ((x - y) / t));
	} else if (y <= 2.1e-49) {
		tmp = x + (z * ((y - x) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y <= -8.8e-61:
		tmp = t_1
	elif y <= 1.38e-124:
		tmp = x + (x * ((z - t) / (t - a)))
	elif y <= 8.4e-75:
		tmp = y + ((z - a) * ((x - y) / t))
	elif y <= 2.1e-49:
		tmp = x + (z * ((y - x) / a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y <= -8.8e-61)
		tmp = t_1;
	elseif (y <= 1.38e-124)
		tmp = Float64(x + Float64(x * Float64(Float64(z - t) / Float64(t - a))));
	elseif (y <= 8.4e-75)
		tmp = Float64(y + Float64(Float64(z - a) * Float64(Float64(x - y) / t)));
	elseif (y <= 2.1e-49)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y <= -8.8e-61)
		tmp = t_1;
	elseif (y <= 1.38e-124)
		tmp = x + (x * ((z - t) / (t - a)));
	elseif (y <= 8.4e-75)
		tmp = y + ((z - a) * ((x - y) / t));
	elseif (y <= 2.1e-49)
		tmp = x + (z * ((y - x) / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e-61], t$95$1, If[LessEqual[y, 1.38e-124], N[(x + N[(x * N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.4e-75], N[(y + N[(N[(z - a), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-49], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.80000000000000035e-61 or 2.0999999999999999e-49 < y

   1. Initial program 65.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 66.3%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* 85.9%

         \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   5. Simplified 85.9%

      \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if -8.80000000000000035e-61 < y < 1.37999999999999995e-124

   1. Initial program 70.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 61.0%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 61.0%

         \[\leadsto x + \color{blue}{\left(-\frac{x \cdot \left(z - t\right)}{a - t}\right)} \]

      2. associate-/l* 70.2%

         \[\leadsto x + \left(-\color{blue}{x \cdot \frac{z - t}{a - t}}\right) \]

      3. distribute-rgt-neg-in 70.2%

         \[\leadsto x + \color{blue}{x \cdot \left(-\frac{z - t}{a - t}\right)} \]

      4. mul-1-neg 70.2%

         \[\leadsto x + x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right)} \]

      5. associate-*r/ 70.2%

         \[\leadsto x + x \cdot \color{blue}{\frac{-1 \cdot \left(z - t\right)}{a - t}} \]

      6. neg-mul-1 70.2%

         \[\leadsto x + x \cdot \frac{\color{blue}{-\left(z - t\right)}}{a - t} \]

      7. neg-sub0 70.2%

         \[\leadsto x + x \cdot \frac{\color{blue}{0 - \left(z - t\right)}}{a - t} \]

      8. sub-neg 70.2%

         \[\leadsto x + x \cdot \frac{0 - \color{blue}{\left(z + \left(-t\right)\right)}}{a - t} \]

      9. +-commutative 70.2%

         \[\leadsto x + x \cdot \frac{0 - \color{blue}{\left(\left(-t\right) + z\right)}}{a - t} \]

      10. associate--r+ 70.2%

          \[\leadsto x + x \cdot \frac{\color{blue}{\left(0 - \left(-t\right)\right) - z}}{a - t} \]

      11. neg-sub0 70.2%

          \[\leadsto x + x \cdot \frac{\color{blue}{\left(-\left(-t\right)\right)} - z}{a - t} \]

      12. remove-double-neg 70.2%

          \[\leadsto x + x \cdot \frac{\color{blue}{t} - z}{a - t} \]

   5. Simplified 70.2%

      \[\leadsto x + \color{blue}{x \cdot \frac{t - z}{a - t}} \]

   if 1.37999999999999995e-124 < y < 8.4000000000000004e-75

   1. Initial program 63.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 51.6%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 51.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 75.5%

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   6. Step-by-step derivation

      1. associate--l+ 75.5%

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- 75.5%

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub 75.5%

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. mul-1-neg 75.5%

         \[\leadsto y + \color{blue}{\left(-\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)} \]

      5. unsub-neg 75.5%

         \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      6. div-sub 75.5%

         \[\leadsto y - \color{blue}{\left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      7. associate-/l* 87.6%

         \[\leadsto y - \left(\color{blue}{z \cdot \frac{y - x}{t}} - \frac{a \cdot \left(y - x\right)}{t}\right) \]

      8. associate-/l* 87.6%

         \[\leadsto y - \left(z \cdot \frac{y - x}{t} - \color{blue}{a \cdot \frac{y - x}{t}}\right) \]

      9. distribute-rgt-out-- 87.6%

         \[\leadsto y - \color{blue}{\frac{y - x}{t} \cdot \left(z - a\right)} \]

   7. Simplified 87.6%

      \[\leadsto \color{blue}{y - \frac{y - x}{t} \cdot \left(z - a\right)} \]

   if 8.4000000000000004e-75 < y < 2.0999999999999999e-49

   1. Initial program 100.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 100.0%

      \[\leadsto x + \color{blue}{\frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]

   5. Simplified 100.0%

      \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-61}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-124}:\\ \;\;\;\;x + x \cdot \frac{z - t}{t - a}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-75}:\\ \;\;\;\;y + \left(z - a\right) \cdot \frac{x - y}{t}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(\left(1 + \frac{t}{a - t}\right) + \frac{z}{t - a}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (<= y -5e-58)
     t_1
     (if (<= y 2.5e-217)
       (* x (+ (+ 1.0 (/ t (- a t))) (/ z (- t a))))
       (if (<= y 1.12e-48) (+ x (/ (* (- y x) (- z t)) (- a t))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y <= -5e-58) {
		tmp = t_1;
	} else if (y <= 2.5e-217) {
		tmp = x * ((1.0 + (t / (a - t))) + (z / (t - a)));
	} else if (y <= 1.12e-48) {
		tmp = x + (((y - x) * (z - t)) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y <= (-5d-58)) then
        tmp = t_1
    else if (y <= 2.5d-217) then
        tmp = x * ((1.0d0 + (t / (a - t))) + (z / (t - a)))
    else if (y <= 1.12d-48) then
        tmp = x + (((y - x) * (z - t)) / (a - t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y <= -5e-58) {
		tmp = t_1;
	} else if (y <= 2.5e-217) {
		tmp = x * ((1.0 + (t / (a - t))) + (z / (t - a)));
	} else if (y <= 1.12e-48) {
		tmp = x + (((y - x) * (z - t)) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y <= -5e-58:
		tmp = t_1
	elif y <= 2.5e-217:
		tmp = x * ((1.0 + (t / (a - t))) + (z / (t - a)))
	elif y <= 1.12e-48:
		tmp = x + (((y - x) * (z - t)) / (a - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y <= -5e-58)
		tmp = t_1;
	elseif (y <= 2.5e-217)
		tmp = Float64(x * Float64(Float64(1.0 + Float64(t / Float64(a - t))) + Float64(z / Float64(t - a))));
	elseif (y <= 1.12e-48)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y <= -5e-58)
		tmp = t_1;
	elseif (y <= 2.5e-217)
		tmp = x * ((1.0 + (t / (a - t))) + (z / (t - a)));
	elseif (y <= 1.12e-48)
		tmp = x + (((y - x) * (z - t)) / (a - t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-58], t$95$1, If[LessEqual[y, 2.5e-217], N[(x * N[(N[(1.0 + N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e-48], N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999977e-58 or 1.11999999999999999e-48 < y

   1. Initial program 65.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 66.3%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* 85.9%

         \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   5. Simplified 85.9%

      \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if -4.99999999999999977e-58 < y < 2.5000000000000001e-217

   1. Initial program 68.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 68.5%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 78.0%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 78.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 82.3%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 82.3%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right)} \]

      2. neg-mul-1 82.3%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{z}{a - t} - \left(1 + \frac{t}{a - t}\right)\right) \]

      3. +-commutative 82.3%

         \[\leadsto \left(-x\right) \cdot \left(\frac{z}{a - t} - \color{blue}{\left(\frac{t}{a - t} + 1\right)}\right) \]

   7. Simplified 82.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{z}{a - t} - \left(\frac{t}{a - t} + 1\right)\right)} \]

   if 2.5000000000000001e-217 < y < 1.11999999999999999e-48

   1. Initial program 76.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-58}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(\left(1 + \frac{t}{a - t}\right) + \frac{z}{t - a}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{a - t}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-227}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-303}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ y (- a t)))))
   (if (<= z -2.5e+72)
     t_1
     (if (<= z -1.75e-227)
       x
       (if (<= z -4.3e-303) y (if (<= z 2.3e+47) x t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / (a - t));
	double tmp;
	if (z <= -2.5e+72) {
		tmp = t_1;
	} else if (z <= -1.75e-227) {
		tmp = x;
	} else if (z <= -4.3e-303) {
		tmp = y;
	} else if (z <= 2.3e+47) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (y / (a - t))
    if (z <= (-2.5d+72)) then
        tmp = t_1
    else if (z <= (-1.75d-227)) then
        tmp = x
    else if (z <= (-4.3d-303)) then
        tmp = y
    else if (z <= 2.3d+47) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (y / (a - t));
	double tmp;
	if (z <= -2.5e+72) {
		tmp = t_1;
	} else if (z <= -1.75e-227) {
		tmp = x;
	} else if (z <= -4.3e-303) {
		tmp = y;
	} else if (z <= 2.3e+47) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (y / (a - t))
	tmp = 0
	if z <= -2.5e+72:
		tmp = t_1
	elif z <= -1.75e-227:
		tmp = x
	elif z <= -4.3e-303:
		tmp = y
	elif z <= 2.3e+47:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(y / Float64(a - t)))
	tmp = 0.0
	if (z <= -2.5e+72)
		tmp = t_1;
	elseif (z <= -1.75e-227)
		tmp = x;
	elseif (z <= -4.3e-303)
		tmp = y;
	elseif (z <= 2.3e+47)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (y / (a - t));
	tmp = 0.0;
	if (z <= -2.5e+72)
		tmp = t_1;
	elseif (z <= -1.75e-227)
		tmp = x;
	elseif (z <= -4.3e-303)
		tmp = y;
	elseif (z <= 2.3e+47)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.5e+72], t$95$1, If[LessEqual[z, -1.75e-227], x, If[LessEqual[z, -4.3e-303], y, If[LessEqual[z, 2.3e+47], x, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.49999999999999996e72 or 2.2999999999999999e47 < z

   1. Initial program 66.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 66.8%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 86.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 86.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 81.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 81.7%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 81.7%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 81.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in z around inf 55.1%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{a - t}} \]

   9. Taylor expanded in y around inf 36.5%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a - t}} \]
   10. Step-by-step derivation

       1. *-commutative 36.5%

          \[\leadsto \frac{\color{blue}{z \cdot y}}{a - t} \]

       2. associate-*r/ 43.1%

          \[\leadsto \color{blue}{z \cdot \frac{y}{a - t}} \]

   11. Simplified 43.1%

       \[\leadsto \color{blue}{z \cdot \frac{y}{a - t}} \]

   if -2.49999999999999996e72 < z < -1.75000000000000005e-227 or -4.29999999999999981e-303 < z < 2.2999999999999999e47

   1. Initial program 72.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 72.2%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 84.9%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 84.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 45.9%

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

   if -1.75000000000000005e-227 < z < -4.29999999999999981e-303

   1. Initial program 38.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 38.4%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 71.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 71.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 49.6%

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

Alternative 8: 36.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{z}{t}\\ \mathbf{if}\;a \leq -1.02 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;y\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ z t))))
   (if (<= a -1.02e-29)
     x
     (if (<= a 1.2e-170)
       t_1
       (if (<= a 1.65e+22) y (if (<= a 1.45e+32) t_1 x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / t);
	double tmp;
	if (a <= -1.02e-29) {
		tmp = x;
	} else if (a <= 1.2e-170) {
		tmp = t_1;
	} else if (a <= 1.65e+22) {
		tmp = y;
	} else if (a <= 1.45e+32) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (z / t)
    if (a <= (-1.02d-29)) then
        tmp = x
    else if (a <= 1.2d-170) then
        tmp = t_1
    else if (a <= 1.65d+22) then
        tmp = y
    else if (a <= 1.45d+32) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (z / t);
	double tmp;
	if (a <= -1.02e-29) {
		tmp = x;
	} else if (a <= 1.2e-170) {
		tmp = t_1;
	} else if (a <= 1.65e+22) {
		tmp = y;
	} else if (a <= 1.45e+32) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * (z / t)
	tmp = 0
	if a <= -1.02e-29:
		tmp = x
	elif a <= 1.2e-170:
		tmp = t_1
	elif a <= 1.65e+22:
		tmp = y
	elif a <= 1.45e+32:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(z / t))
	tmp = 0.0
	if (a <= -1.02e-29)
		tmp = x;
	elseif (a <= 1.2e-170)
		tmp = t_1;
	elseif (a <= 1.65e+22)
		tmp = y;
	elseif (a <= 1.45e+32)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (z / t);
	tmp = 0.0;
	if (a <= -1.02e-29)
		tmp = x;
	elseif (a <= 1.2e-170)
		tmp = t_1;
	elseif (a <= 1.65e+22)
		tmp = y;
	elseif (a <= 1.45e+32)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.02e-29], x, If[LessEqual[a, 1.2e-170], t$95$1, If[LessEqual[a, 1.65e+22], y, If[LessEqual[a, 1.45e+32], t$95$1, x]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999994e-29 or 1.45000000000000001e32 < a

   1. Initial program 67.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 67.0%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 93.6%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 93.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 46.6%

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

   if -1.01999999999999994e-29 < a < 1.2e-170 or 1.6499999999999999e22 < a < 1.45000000000000001e32

   1. Initial program 68.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 68.0%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 75.0%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 75.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 71.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 71.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 71.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 71.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in z around inf 48.6%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{a - t}} \]

   9. Taylor expanded in a around 0 38.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{t}} \]
   10. Step-by-step derivation

       1. mul-1-neg 38.6%

          \[\leadsto \color{blue}{-\frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{t}} \]

       2. associate-/l* 36.4%

          \[\leadsto -\color{blue}{y \cdot \frac{z \cdot \left(1 - \frac{x}{y}\right)}{t}} \]

       3. distribute-rgt-neg-in 36.4%

          \[\leadsto \color{blue}{y \cdot \left(-\frac{z \cdot \left(1 - \frac{x}{y}\right)}{t}\right)} \]

   11. Simplified 36.4%

       \[\leadsto \color{blue}{y \cdot \left(-\frac{z \cdot \left(1 - \frac{x}{y}\right)}{t}\right)} \]

   12. Taylor expanded in y around 0 35.3%

       \[\leadsto \color{blue}{\frac{x \cdot z}{t}} \]
   13. Step-by-step derivation

       1. associate-/l* 39.7%

          \[\leadsto \color{blue}{x \cdot \frac{z}{t}} \]

   14. Simplified 39.7%

       \[\leadsto \color{blue}{x \cdot \frac{z}{t}} \]

   if 1.2e-170 < a < 1.6499999999999999e22

   1. Initial program 69.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 69.0%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 73.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 73.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 29.8%

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

Alternative 9: 71.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-60} \lor \neg \left(y \leq 1.12 \cdot 10^{-95}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{z - t}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -1.65e-60) (not (<= y 1.12e-95)))
   (+ x (* y (/ (- z t) (- a t))))
   (+ x (* x (/ (- z t) (- t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.65e-60) || !(y <= 1.12e-95)) {
		tmp = x + (y * ((z - t) / (a - t)));
	} else {
		tmp = x + (x * ((z - t) / (t - a)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y <= (-1.65d-60)) .or. (.not. (y <= 1.12d-95))) then
        tmp = x + (y * ((z - t) / (a - t)))
    else
        tmp = x + (x * ((z - t) / (t - a)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.65e-60) || !(y <= 1.12e-95)) {
		tmp = x + (y * ((z - t) / (a - t)));
	} else {
		tmp = x + (x * ((z - t) / (t - a)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -1.65e-60) or not (y <= 1.12e-95):
		tmp = x + (y * ((z - t) / (a - t)))
	else:
		tmp = x + (x * ((z - t) / (t - a)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -1.65e-60) || !(y <= 1.12e-95))
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))));
	else
		tmp = Float64(x + Float64(x * Float64(Float64(z - t) / Float64(t - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -1.65e-60) || ~((y <= 1.12e-95)))
		tmp = x + (y * ((z - t) / (a - t)));
	else
		tmp = x + (x * ((z - t) / (t - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -1.65e-60], N[Not[LessEqual[y, 1.12e-95]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6499999999999999e-60 or 1.12000000000000006e-95 < y

   1. Initial program 65.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 66.2%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. associate-/l* 84.3%

         \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   5. Simplified 84.3%

      \[\leadsto x + \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if -1.6499999999999999e-60 < y < 1.12000000000000006e-95

   1. Initial program 70.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 60.5%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x \cdot \left(z - t\right)}{a - t}} \]
   4. Step-by-step derivation

      1. mul-1-neg 60.5%

         \[\leadsto x + \color{blue}{\left(-\frac{x \cdot \left(z - t\right)}{a - t}\right)} \]

      2. associate-/l* 69.2%

         \[\leadsto x + \left(-\color{blue}{x \cdot \frac{z - t}{a - t}}\right) \]

      3. distribute-rgt-neg-in 69.2%

         \[\leadsto x + \color{blue}{x \cdot \left(-\frac{z - t}{a - t}\right)} \]

      4. mul-1-neg 69.2%

         \[\leadsto x + x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right)} \]

      5. associate-*r/ 69.2%

         \[\leadsto x + x \cdot \color{blue}{\frac{-1 \cdot \left(z - t\right)}{a - t}} \]

      6. neg-mul-1 69.2%

         \[\leadsto x + x \cdot \frac{\color{blue}{-\left(z - t\right)}}{a - t} \]

      7. neg-sub0 69.2%

         \[\leadsto x + x \cdot \frac{\color{blue}{0 - \left(z - t\right)}}{a - t} \]

      8. sub-neg 69.2%

         \[\leadsto x + x \cdot \frac{0 - \color{blue}{\left(z + \left(-t\right)\right)}}{a - t} \]

      9. +-commutative 69.2%

         \[\leadsto x + x \cdot \frac{0 - \color{blue}{\left(\left(-t\right) + z\right)}}{a - t} \]

      10. associate--r+ 69.2%

          \[\leadsto x + x \cdot \frac{\color{blue}{\left(0 - \left(-t\right)\right) - z}}{a - t} \]

      11. neg-sub0 69.2%

          \[\leadsto x + x \cdot \frac{\color{blue}{\left(-\left(-t\right)\right)} - z}{a - t} \]

      12. remove-double-neg 69.2%

          \[\leadsto x + x \cdot \frac{\color{blue}{t} - z}{a - t} \]

   5. Simplified 69.2%

      \[\leadsto x + \color{blue}{x \cdot \frac{t - z}{a - t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-60} \lor \neg \left(y \leq 1.12 \cdot 10^{-95}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{z - t}{t - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-23} \lor \neg \left(t \leq 1.7 \cdot 10^{+86}\right):\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2e-23) (not (<= t 1.7e+86)))
   (+ y (* x (/ (- z a) t)))
   (+ x (* z (/ (- y x) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2e-23) || !(t <= 1.7e+86)) {
		tmp = y + (x * ((z - a) / t));
	} else {
		tmp = x + (z * ((y - x) / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-2d-23)) .or. (.not. (t <= 1.7d+86))) then
        tmp = y + (x * ((z - a) / t))
    else
        tmp = x + (z * ((y - x) / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2e-23) || !(t <= 1.7e+86)) {
		tmp = y + (x * ((z - a) / t));
	} else {
		tmp = x + (z * ((y - x) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2e-23) or not (t <= 1.7e+86):
		tmp = y + (x * ((z - a) / t))
	else:
		tmp = x + (z * ((y - x) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2e-23) || !(t <= 1.7e+86))
		tmp = Float64(y + Float64(x * Float64(Float64(z - a) / t)));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2e-23) || ~((t <= 1.7e+86)))
		tmp = y + (x * ((z - a) / t));
	else
		tmp = x + (z * ((y - x) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2e-23], N[Not[LessEqual[t, 1.7e+86]], $MachinePrecision]], N[(y + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999992e-23 or 1.6999999999999999e86 < t

   1. Initial program 39.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 39.4%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 72.5%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 72.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 58.4%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in y around 0 58.3%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 58.3%

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}{t}} \]

      2. mul-1-neg 58.3%

         \[\leadsto y + \frac{\color{blue}{-\left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}}{t} \]

      3. associate-*r* 58.3%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-1 \cdot a\right) \cdot x}\right)}{t} \]

      4. mul-1-neg 58.3%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-a\right)} \cdot x\right)}{t} \]

      5. cancel-sign-sub 58.3%

         \[\leadsto y + \frac{-\color{blue}{\left(-1 \cdot \left(x \cdot z\right) + a \cdot x\right)}}{t} \]

      6. mul-1-neg 58.3%

         \[\leadsto y + \frac{-\left(\color{blue}{\left(-x \cdot z\right)} + a \cdot x\right)}{t} \]

      7. distribute-rgt-neg-in 58.3%

         \[\leadsto y + \frac{-\left(\color{blue}{x \cdot \left(-z\right)} + a \cdot x\right)}{t} \]

      8. mul-1-neg 58.3%

         \[\leadsto y + \frac{-\left(x \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot x\right)}{t} \]

      9. *-commutative 58.3%

         \[\leadsto y + \frac{-\left(x \cdot \left(-1 \cdot z\right) + \color{blue}{x \cdot a}\right)}{t} \]

      10. distribute-lft-in 58.3%

          \[\leadsto y + \frac{-\color{blue}{x \cdot \left(-1 \cdot z + a\right)}}{t} \]

      11. *-lft-identity 58.3%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{1 \cdot a}\right)}{t} \]

      12. metadata-eval 58.3%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{\left(--1\right)} \cdot a\right)}{t} \]

      13. cancel-sign-sub-inv 58.3%

          \[\leadsto y + \frac{-x \cdot \color{blue}{\left(-1 \cdot z - -1 \cdot a\right)}}{t} \]

      14. distribute-neg-frac 58.3%

          \[\leadsto y + \color{blue}{\left(-\frac{x \cdot \left(-1 \cdot z - -1 \cdot a\right)}{t}\right)} \]

      15. associate-/l* 65.2%

          \[\leadsto y + \left(-\color{blue}{x \cdot \frac{-1 \cdot z - -1 \cdot a}{t}}\right) \]

      16. distribute-lft-out-- 65.2%

          \[\leadsto y + \left(-x \cdot \frac{\color{blue}{-1 \cdot \left(z - a\right)}}{t}\right) \]

      17. associate-*r/ 65.2%

          \[\leadsto y + \left(-x \cdot \color{blue}{\left(-1 \cdot \frac{z - a}{t}\right)}\right) \]

   8. Simplified 65.2%

      \[\leadsto y + \color{blue}{x \cdot \frac{z - a}{t}} \]

   if -1.99999999999999992e-23 < t < 1.6999999999999999e86

   1. Initial program 88.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 68.6%

      \[\leadsto x + \color{blue}{\frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. associate-/l* 75.5%

         \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]

   5. Simplified 75.5%

      \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-23} \lor \neg \left(t \leq 1.7 \cdot 10^{+86}\right):\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+64} \lor \neg \left(t \leq 1.96 \cdot 10^{-44}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -5.6e+64) (not (<= t 1.96e-44)))
   (* y (/ (- z t) (- a t)))
   (+ x (* z (/ (- y x) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.6e+64) || !(t <= 1.96e-44)) {
		tmp = y * ((z - t) / (a - t));
	} else {
		tmp = x + (z * ((y - x) / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-5.6d+64)) .or. (.not. (t <= 1.96d-44))) then
        tmp = y * ((z - t) / (a - t))
    else
        tmp = x + (z * ((y - x) / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.6e+64) || !(t <= 1.96e-44)) {
		tmp = y * ((z - t) / (a - t));
	} else {
		tmp = x + (z * ((y - x) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -5.6e+64) or not (t <= 1.96e-44):
		tmp = y * ((z - t) / (a - t))
	else:
		tmp = x + (z * ((y - x) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -5.6e+64) || !(t <= 1.96e-44))
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -5.6e+64) || ~((t <= 1.96e-44)))
		tmp = y * ((z - t) / (a - t));
	else
		tmp = x + (z * ((y - x) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -5.6e+64], N[Not[LessEqual[t, 1.96e-44]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.60000000000000047e64 or 1.9599999999999999e-44 < t

   1. Initial program 45.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 45.7%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 77.9%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 78.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 72.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 72.3%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 72.3%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 72.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in y around inf 57.2%

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{a - t} - \frac{t}{a - t}\right)} \]
   9. Step-by-step derivation

      1. div-sub 57.2%

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

   10. Simplified 57.2%

       \[\leadsto \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if -5.60000000000000047e64 < t < 1.9599999999999999e-44

   1. Initial program 86.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 67.9%

      \[\leadsto x + \color{blue}{\frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. associate-/l* 75.9%

         \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]

   5. Simplified 75.9%

      \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+64} \lor \neg \left(t \leq 1.96 \cdot 10^{-44}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-23}:\\ \;\;\;\;y + x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-44}:\\ \;\;\;\;x + z \cdot \frac{y - x}{a}\\ \mathbf{else}:\\ \;\;\;\;y + z \cdot \frac{x - y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.4e-23)
   (+ y (* x (/ (- z a) t)))
   (if (<= t 1.3e-44) (+ x (* z (/ (- y x) a))) (+ y (* z (/ (- x y) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.4e-23) {
		tmp = y + (x * ((z - a) / t));
	} else if (t <= 1.3e-44) {
		tmp = x + (z * ((y - x) / a));
	} else {
		tmp = y + (z * ((x - y) / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.4d-23)) then
        tmp = y + (x * ((z - a) / t))
    else if (t <= 1.3d-44) then
        tmp = x + (z * ((y - x) / a))
    else
        tmp = y + (z * ((x - y) / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.4e-23) {
		tmp = y + (x * ((z - a) / t));
	} else if (t <= 1.3e-44) {
		tmp = x + (z * ((y - x) / a));
	} else {
		tmp = y + (z * ((x - y) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.4e-23:
		tmp = y + (x * ((z - a) / t))
	elif t <= 1.3e-44:
		tmp = x + (z * ((y - x) / a))
	else:
		tmp = y + (z * ((x - y) / t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.4e-23)
		tmp = Float64(y + Float64(x * Float64(Float64(z - a) / t)));
	elseif (t <= 1.3e-44)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / a)));
	else
		tmp = Float64(y + Float64(z * Float64(Float64(x - y) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.4e-23)
		tmp = y + (x * ((z - a) / t));
	elseif (t <= 1.3e-44)
		tmp = x + (z * ((y - x) / a));
	else
		tmp = y + (z * ((x - y) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.4e-23], N[(y + N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-44], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.3999999999999999e-23

   1. Initial program 45.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 45.8%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 71.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 71.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 63.8%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in y around 0 60.3%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 60.3%

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}{t}} \]

      2. mul-1-neg 60.3%

         \[\leadsto y + \frac{\color{blue}{-\left(-1 \cdot \left(x \cdot z\right) - -1 \cdot \left(a \cdot x\right)\right)}}{t} \]

      3. associate-*r* 60.3%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-1 \cdot a\right) \cdot x}\right)}{t} \]

      4. mul-1-neg 60.3%

         \[\leadsto y + \frac{-\left(-1 \cdot \left(x \cdot z\right) - \color{blue}{\left(-a\right)} \cdot x\right)}{t} \]

      5. cancel-sign-sub 60.3%

         \[\leadsto y + \frac{-\color{blue}{\left(-1 \cdot \left(x \cdot z\right) + a \cdot x\right)}}{t} \]

      6. mul-1-neg 60.3%

         \[\leadsto y + \frac{-\left(\color{blue}{\left(-x \cdot z\right)} + a \cdot x\right)}{t} \]

      7. distribute-rgt-neg-in 60.3%

         \[\leadsto y + \frac{-\left(\color{blue}{x \cdot \left(-z\right)} + a \cdot x\right)}{t} \]

      8. mul-1-neg 60.3%

         \[\leadsto y + \frac{-\left(x \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot x\right)}{t} \]

      9. *-commutative 60.3%

         \[\leadsto y + \frac{-\left(x \cdot \left(-1 \cdot z\right) + \color{blue}{x \cdot a}\right)}{t} \]

      10. distribute-lft-in 60.3%

          \[\leadsto y + \frac{-\color{blue}{x \cdot \left(-1 \cdot z + a\right)}}{t} \]

      11. *-lft-identity 60.3%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{1 \cdot a}\right)}{t} \]

      12. metadata-eval 60.3%

          \[\leadsto y + \frac{-x \cdot \left(-1 \cdot z + \color{blue}{\left(--1\right)} \cdot a\right)}{t} \]

      13. cancel-sign-sub-inv 60.3%

          \[\leadsto y + \frac{-x \cdot \color{blue}{\left(-1 \cdot z - -1 \cdot a\right)}}{t} \]

      14. distribute-neg-frac 60.3%

          \[\leadsto y + \color{blue}{\left(-\frac{x \cdot \left(-1 \cdot z - -1 \cdot a\right)}{t}\right)} \]

      15. associate-/l* 61.9%

          \[\leadsto y + \left(-\color{blue}{x \cdot \frac{-1 \cdot z - -1 \cdot a}{t}}\right) \]

      16. distribute-lft-out-- 61.9%

          \[\leadsto y + \left(-x \cdot \frac{\color{blue}{-1 \cdot \left(z - a\right)}}{t}\right) \]

      17. associate-*r/ 61.9%

          \[\leadsto y + \left(-x \cdot \color{blue}{\left(-1 \cdot \frac{z - a}{t}\right)}\right) \]

   8. Simplified 61.9%

      \[\leadsto y + \color{blue}{x \cdot \frac{z - a}{t}} \]

   if -1.3999999999999999e-23 < t < 1.2999999999999999e-44

   1. Initial program 90.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 74.0%

      \[\leadsto x + \color{blue}{\frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. associate-/l* 81.5%

         \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]

   5. Simplified 81.5%

      \[\leadsto x + \color{blue}{z \cdot \frac{y - x}{a}} \]

   if 1.2999999999999999e-44 < t

   1. Initial program 47.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 47.1%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 80.0%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 80.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 52.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   6. Taylor expanded in z around inf 56.1%

      \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right)}{t}} \]
   7. Step-by-step derivation

      1. mul-1-neg 56.1%

         \[\leadsto y + \color{blue}{\left(-\frac{z \cdot \left(y - x\right)}{t}\right)} \]

      2. associate-/l* 65.4%

         \[\leadsto y + \left(-\color{blue}{z \cdot \frac{y - x}{t}}\right) \]

      3. distribute-rgt-neg-in 65.4%

         \[\leadsto y + \color{blue}{z \cdot \left(-\frac{y - x}{t}\right)} \]

      4. sub-neg 65.4%

         \[\leadsto y + z \cdot \left(-\frac{\color{blue}{y + \left(-x\right)}}{t}\right) \]

      5. mul-1-neg 65.4%

         \[\leadsto y + z \cdot \left(-\frac{y + \color{blue}{-1 \cdot x}}{t}\right) \]

      6. +-commutative 65.4%

         \[\leadsto y + z \cdot \left(-\frac{\color{blue}{-1 \cdot x + y}}{t}\right) \]

      7. remove-double-neg 65.4%

         \[\leadsto y + z \cdot \left(-\frac{-1 \cdot x + \color{blue}{\left(-\left(-y\right)\right)}}{t}\right) \]

      8. neg-mul-1 65.4%

         \[\leadsto y + z \cdot \left(-\frac{-1 \cdot x + \left(-\color{blue}{-1 \cdot y}\right)}{t}\right) \]

      9. sub-neg 65.4%

         \[\leadsto y + z \cdot \left(-\frac{\color{blue}{-1 \cdot x - -1 \cdot y}}{t}\right) \]

      10. distribute-lft-out-- 65.4%

          \[\leadsto y + z \cdot \left(-\frac{\color{blue}{-1 \cdot \left(x - y\right)}}{t}\right) \]

      11. mul-1-neg 65.4%

          \[\leadsto y + z \cdot \left(-\frac{\color{blue}{-\left(x - y\right)}}{t}\right) \]

      12. distribute-neg-frac 65.4%

          \[\leadsto y + z \cdot \left(-\color{blue}{\left(-\frac{x - y}{t}\right)}\right) \]

      13. distribute-neg-frac2 65.4%

          \[\leadsto y + z \cdot \left(-\color{blue}{\frac{x - y}{-t}}\right) \]

      14. distribute-frac-neg2 65.4%

          \[\leadsto y + z \cdot \color{blue}{\frac{x - y}{-\left(-t\right)}} \]

      15. remove-double-neg 65.4%

          \[\leadsto y + z \cdot \frac{x - y}{\color{blue}{t}} \]

   8. Simplified 65.4%

      \[\leadsto y + \color{blue}{z \cdot \frac{x - y}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19000000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -19000000000000.0)
   x
   (if (<= x 1.55e+25) (* y (/ (- z t) (- a t))) (/ (* x z) (- t a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -19000000000000.0) {
		tmp = x;
	} else if (x <= 1.55e+25) {
		tmp = y * ((z - t) / (a - t));
	} else {
		tmp = (x * z) / (t - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-19000000000000.0d0)) then
        tmp = x
    else if (x <= 1.55d+25) then
        tmp = y * ((z - t) / (a - t))
    else
        tmp = (x * z) / (t - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -19000000000000.0) {
		tmp = x;
	} else if (x <= 1.55e+25) {
		tmp = y * ((z - t) / (a - t));
	} else {
		tmp = (x * z) / (t - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -19000000000000.0:
		tmp = x
	elif x <= 1.55e+25:
		tmp = y * ((z - t) / (a - t))
	else:
		tmp = (x * z) / (t - a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -19000000000000.0)
		tmp = x;
	elseif (x <= 1.55e+25)
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	else
		tmp = Float64(Float64(x * z) / Float64(t - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -19000000000000.0)
		tmp = x;
	elseif (x <= 1.55e+25)
		tmp = y * ((z - t) / (a - t));
	else
		tmp = (x * z) / (t - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -19000000000000.0], x, If[LessEqual[x, 1.55e+25], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9e13

   1. Initial program 59.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 59.9%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 84.2%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 84.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 48.3%

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

   if -1.9e13 < x < 1.5499999999999999e25

   1. Initial program 75.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 75.1%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 87.4%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 87.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 87.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 87.4%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 87.4%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 87.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in y around inf 67.4%

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{a - t} - \frac{t}{a - t}\right)} \]
   9. Step-by-step derivation

      1. div-sub 67.4%

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

   10. Simplified 67.4%

       \[\leadsto \color{blue}{y \cdot \frac{z - t}{a - t}} \]

   if 1.5499999999999999e25 < x

   1. Initial program 57.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 57.9%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 79.4%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 79.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 60.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 60.9%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 60.9%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 60.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in z around inf 35.9%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{a - t}} \]

   9. Taylor expanded in y around 0 45.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{a - t}} \]
   10. Step-by-step derivation

       1. associate-*r/ 45.5%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{a - t}} \]

       2. mul-1-neg 45.5%

          \[\leadsto \frac{\color{blue}{-x \cdot z}}{a - t} \]

       3. distribute-rgt-neg-in 45.5%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-z\right)}}{a - t} \]

   11. Simplified 45.5%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-z\right)}{a - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19000000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{t - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-72}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -8.6e-51)
   x
   (if (<= x 4.8e-72) (* z (/ y (- a t))) (/ (* x z) (- t a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.6e-51) {
		tmp = x;
	} else if (x <= 4.8e-72) {
		tmp = z * (y / (a - t));
	} else {
		tmp = (x * z) / (t - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-8.6d-51)) then
        tmp = x
    else if (x <= 4.8d-72) then
        tmp = z * (y / (a - t))
    else
        tmp = (x * z) / (t - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.6e-51) {
		tmp = x;
	} else if (x <= 4.8e-72) {
		tmp = z * (y / (a - t));
	} else {
		tmp = (x * z) / (t - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -8.6e-51:
		tmp = x
	elif x <= 4.8e-72:
		tmp = z * (y / (a - t))
	else:
		tmp = (x * z) / (t - a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -8.6e-51)
		tmp = x;
	elseif (x <= 4.8e-72)
		tmp = Float64(z * Float64(y / Float64(a - t)));
	else
		tmp = Float64(Float64(x * z) / Float64(t - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -8.6e-51)
		tmp = x;
	elseif (x <= 4.8e-72)
		tmp = z * (y / (a - t));
	else
		tmp = (x * z) / (t - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -8.6e-51], x, If[LessEqual[x, 4.8e-72], N[(z * N[(y / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5999999999999995e-51

   1. Initial program 60.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 60.0%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 82.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 82.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 44.8%

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

   if -8.5999999999999995e-51 < x < 4.8e-72

   1. Initial program 78.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 78.0%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 87.6%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 87.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 87.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 87.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 87.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 87.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in z around inf 42.0%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{a - t}} \]

   9. Taylor expanded in y around inf 37.9%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a - t}} \]
   10. Step-by-step derivation

       1. *-commutative 37.9%

          \[\leadsto \frac{\color{blue}{z \cdot y}}{a - t} \]

       2. associate-*r/ 42.3%

          \[\leadsto \color{blue}{z \cdot \frac{y}{a - t}} \]

   11. Simplified 42.3%

       \[\leadsto \color{blue}{z \cdot \frac{y}{a - t}} \]

   if 4.8e-72 < x

   1. Initial program 59.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 59.9%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 82.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 82.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 68.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -1 \cdot \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 68.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \left(1 + \color{blue}{\left(-\frac{x}{y}\right)}\right), \frac{z - t}{a - t}, x\right) \]

      2. unsub-neg 68.5%

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   7. Simplified 68.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(1 - \frac{x}{y}\right)}, \frac{z - t}{a - t}, x\right) \]

   8. Taylor expanded in z around inf 29.4%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot \left(1 - \frac{x}{y}\right)\right)}{a - t}} \]

   9. Taylor expanded in y around 0 39.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{a - t}} \]
   10. Step-by-step derivation

       1. associate-*r/ 39.7%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot z\right)}{a - t}} \]

       2. mul-1-neg 39.7%

          \[\leadsto \frac{\color{blue}{-x \cdot z}}{a - t} \]

       3. distribute-rgt-neg-in 39.7%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-z\right)}}{a - t} \]

   11. Simplified 39.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-z\right)}{a - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-72}:\\ \;\;\;\;z \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{t - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.15 \cdot 10^{+47}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.9e-62) x (if (<= a 3.15e+47) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.9e-62) {
		tmp = x;
	} else if (a <= 3.15e+47) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.9d-62)) then
        tmp = x
    else if (a <= 3.15d+47) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.9e-62) {
		tmp = x;
	} else if (a <= 3.15e+47) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.9e-62:
		tmp = x
	elif a <= 3.15e+47:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.9e-62)
		tmp = x;
	elseif (a <= 3.15e+47)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.9e-62)
		tmp = x;
	elseif (a <= 3.15e+47)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.9e-62], x, If[LessEqual[a, 3.15e+47], y, x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.90000000000000003e-62 or 3.15000000000000002e47 < a

   1. Initial program 67.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 67.9%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 93.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 93.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 46.5%

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

   if -1.90000000000000003e-62 < a < 3.15000000000000002e47

   1. Initial program 67.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. +-commutative 67.2%

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      2. associate-/l* 73.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      3. fma-define 73.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 29.2%

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

Alternative 16: 24.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x;
}

Python

def code(x, y, z, t, a):
	return x

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 67.6%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
2. Step-by-step derivation

   1. +-commutative 67.6%

      \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

   2. associate-/l* 84.8%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

   3. fma-define 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

3. Simplified 84.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around inf 28.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024111 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))