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

Percentage Accurate: 68.0% → 83.1%

Time: 9.9s

Alternatives: 18

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 - z) * (t - x)) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 - z) * (t - x)) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-115}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+185}:\\ \;\;\;\;x + \left(-t\right) \cdot \left(\frac{y - z}{a - z} \cdot \left(\frac{x}{t} - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- (- t x)) z) (- y a) t)))
   (if (<= z -7.8e+147)
     t_1
     (if (<= z 1.25e-115)
       (+ x (/ (* (- y z) (- t x)) (- a z)))
       (if (<= z 2.6e+185)
         (+ x (* (- t) (* (/ (- y z) (- a z)) (- (/ x t) 1.0))))
         t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-(t - x) / z), (y - a), t);
	double tmp;
	if (z <= -7.8e+147) {
		tmp = t_1;
	} else if (z <= 1.25e-115) {
		tmp = x + (((y - z) * (t - x)) / (a - z));
	} else if (z <= 2.6e+185) {
		tmp = x + (-t * (((y - z) / (a - z)) * ((x / t) - 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -7.8e+147)
		tmp = t_1;
	elseif (z <= 1.25e-115)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)));
	elseif (z <= 2.6e+185)
		tmp = Float64(x + Float64(Float64(-t) * Float64(Float64(Float64(y - z) / Float64(a - z)) * Float64(Float64(x / t) - 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.8e+147], t$95$1, If[LessEqual[z, 1.25e-115], N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e+185], N[(x + N[((-t) * N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(N[(x / t), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.80000000000000033e147 or 2.60000000000000001e185 < z

   1. Initial program 28.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out-- N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

   if -7.80000000000000033e147 < z < 1.2500000000000001e-115

   1. Initial program 89.2%

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

   if 1.2500000000000001e-115 < z < 2.60000000000000001e185

   1. Initial program 69.9%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 81.7%

      \[\leadsto x + \color{blue}{\left(-t\right) \cdot \left(\frac{y - z}{a - z} \cdot \left(\frac{x}{t} - 1\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) \cdot \frac{y - z}{z}\\ t_2 := \left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- t) (/ (- y z) z))) (t_2 (* (- t x) (/ y (- a z)))))
   (if (<= z -8.2e+148)
     t_1
     (if (<= z -6.8e-89)
       (fma (- x t) (/ z (- a z)) x)
       (if (<= z -5.8e-161)
         t_2
         (if (<= z 2.3e-12)
           (fma (- y z) (/ (- t x) a) x)
           (if (<= z 1.35e+151) t_2 t_1)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * ((y - z) / z);
	double t_2 = (t - x) * (y / (a - z));
	double tmp;
	if (z <= -8.2e+148) {
		tmp = t_1;
	} else if (z <= -6.8e-89) {
		tmp = fma((x - t), (z / (a - z)), x);
	} else if (z <= -5.8e-161) {
		tmp = t_2;
	} else if (z <= 2.3e-12) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else if (z <= 1.35e+151) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) * Float64(Float64(y - z) / z))
	t_2 = Float64(Float64(t - x) * Float64(y / Float64(a - z)))
	tmp = 0.0
	if (z <= -8.2e+148)
		tmp = t_1;
	elseif (z <= -6.8e-89)
		tmp = fma(Float64(x - t), Float64(z / Float64(a - z)), x);
	elseif (z <= -5.8e-161)
		tmp = t_2;
	elseif (z <= 2.3e-12)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	elseif (z <= 1.35e+151)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e+148], t$95$1, If[LessEqual[z, -6.8e-89], N[(N[(x - t), $MachinePrecision] * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -5.8e-161], t$95$2, If[LessEqual[z, 2.3e-12], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.35e+151], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.1999999999999996e148 or 1.3500000000000001e151 < z

   1. Initial program 31.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.9%

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

   if -8.1999999999999996e148 < z < -6.8000000000000001e-89

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.0%

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

   if -6.8000000000000001e-89 < z < -5.8e-161 or 2.29999999999999989e-12 < z < 1.3500000000000001e151

   1. Initial program 68.3%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if -5.8e-161 < z < 2.29999999999999989e-12

   1. Initial program 95.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 3: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-161}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- (- t x)) z) (- y a) t)))
   (if (<= z -1.05e+132)
     t_1
     (if (<= z -6.8e-89)
       (fma (- x t) (/ z (- a z)) x)
       (if (<= z -5.8e-161)
         (* (- t x) (/ y (- a z)))
         (if (<= z 7.5e-13) (fma (- y z) (/ (- t x) a) x) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-(t - x) / z), (y - a), t);
	double tmp;
	if (z <= -1.05e+132) {
		tmp = t_1;
	} else if (z <= -6.8e-89) {
		tmp = fma((x - t), (z / (a - z)), x);
	} else if (z <= -5.8e-161) {
		tmp = (t - x) * (y / (a - z));
	} else if (z <= 7.5e-13) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -1.05e+132)
		tmp = t_1;
	elseif (z <= -6.8e-89)
		tmp = fma(Float64(x - t), Float64(z / Float64(a - z)), x);
	elseif (z <= -5.8e-161)
		tmp = Float64(Float64(t - x) * Float64(y / Float64(a - z)));
	elseif (z <= 7.5e-13)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.05e+132], t$95$1, If[LessEqual[z, -6.8e-89], N[(N[(x - t), $MachinePrecision] * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -5.8e-161], N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e-13], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.04999999999999997e132 or 7.5000000000000004e-13 < z

   1. Initial program 41.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out-- N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 81.5%

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

   if -1.04999999999999997e132 < z < -6.8000000000000001e-89

   1. Initial program 73.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.1%

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

   if -6.8000000000000001e-89 < z < -5.8e-161

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if -5.8e-161 < z < 7.5000000000000004e-13

   1. Initial program 95.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 4: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-161}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- (- t x)) z) (- y a) t)))
   (if (<= z -7.8e+147)
     t_1
     (if (<= z -4.8e-161)
       (+ x (/ (* (- y z) t) (- a z)))
       (if (<= z 7.5e-13) (fma (- y z) (/ (- t x) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((-(t - x) / z), (y - a), t);
	double tmp;
	if (z <= -7.8e+147) {
		tmp = t_1;
	} else if (z <= -4.8e-161) {
		tmp = x + (((y - z) * t) / (a - z));
	} else if (z <= 7.5e-13) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -7.8e+147)
		tmp = t_1;
	elseif (z <= -4.8e-161)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	elseif (z <= 7.5e-13)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.8e+147], t$95$1, If[LessEqual[z, -4.8e-161], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e-13], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.80000000000000033e147 or 7.5000000000000004e-13 < z

   1. Initial program 39.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out-- N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 82.0%

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

   if -7.80000000000000033e147 < z < -4.79999999999999998e-161

   1. Initial program 78.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -4.79999999999999998e-161 < z < 7.5000000000000004e-13

   1. Initial program 95.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 87.0

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

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+147} \lor \neg \left(z \leq 2.4 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.8e+147) (not (<= z 2.4e+95)))
   (fma (/ (- (- t x)) z) (- y a) t)
   (+ x (/ (* (- y z) (- t x)) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.8e+147) || !(z <= 2.4e+95)) {
		tmp = fma((-(t - x) / z), (y - a), t);
	} else {
		tmp = x + (((y - z) * (t - x)) / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.8e+147) || !(z <= 2.4e+95))
		tmp = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t);
	else
		tmp = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.8e+147], N[Not[LessEqual[z, 2.4e+95]], $MachinePrecision]], N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.80000000000000033e147 or 2.4e95 < z

   1. Initial program 31.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out-- N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 86.4%

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

   if -7.80000000000000033e147 < z < 2.4e95

   1. Initial program 86.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+147} \lor \neg \left(z \leq 2.4 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+41}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+151}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e+41)
   (* (- t) (/ z (- a z)))
   (if (<= z 1.25e-13)
     (fma (/ (- t x) a) y x)
     (if (<= z 1.35e+151) (* (- t x) (/ y (- a z))) (* (- t) (/ (- y z) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+41) {
		tmp = -t * (z / (a - z));
	} else if (z <= 1.25e-13) {
		tmp = fma(((t - x) / a), y, x);
	} else if (z <= 1.35e+151) {
		tmp = (t - x) * (y / (a - z));
	} else {
		tmp = -t * ((y - z) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e+41)
		tmp = Float64(Float64(-t) * Float64(z / Float64(a - z)));
	elseif (z <= 1.25e-13)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	elseif (z <= 1.35e+151)
		tmp = Float64(Float64(t - x) * Float64(y / Float64(a - z)));
	else
		tmp = Float64(Float64(-t) * Float64(Float64(y - z) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e+41], N[((-t) * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e-13], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 1.35e+151], N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t) * N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.7999999999999994e41

   1. Initial program 37.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

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

   if -7.7999999999999994e41 < z < 1.24999999999999997e-13

   1. Initial program 91.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if 1.24999999999999997e-13 < z < 1.3500000000000001e151

   1. Initial program 61.3%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if 1.3500000000000001e151 < z

   1. Initial program 37.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.1%

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

Alternative 7: 42.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{a}, z, x\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-90}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) a) z x)))
   (if (<= a -7e+93)
     t_1
     (if (<= a -3e-90)
       (* t (/ y (- a z)))
       (if (<= a 5.2e+20) (fma a (/ t z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / a), z, x);
	double tmp;
	if (a <= -7e+93) {
		tmp = t_1;
	} else if (a <= -3e-90) {
		tmp = t * (y / (a - z));
	} else if (a <= 5.2e+20) {
		tmp = fma(a, (t / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / a), z, x)
	tmp = 0.0
	if (a <= -7e+93)
		tmp = t_1;
	elseif (a <= -3e-90)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (a <= 5.2e+20)
		tmp = fma(a, Float64(t / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[a, -7e+93], t$95$1, If[LessEqual[a, -3e-90], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.2e+20], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.99999999999999996e93 or 5.2e20 < a

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 60.9%

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

   if -6.99999999999999996e93 < a < -3.0000000000000002e-90

   1. Initial program 76.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 35.9%

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

   if -3.0000000000000002e-90 < a < 5.2e20

   1. Initial program 60.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(a, \frac{t}{\color{blue}{z}}, t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 42.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-90}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ z a) x)))
   (if (<= a -7e+93)
     t_1
     (if (<= a -3e-90)
       (* t (/ y (- a z)))
       (if (<= a 5.2e+20) (fma a (/ t z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (z / a), x);
	double tmp;
	if (a <= -7e+93) {
		tmp = t_1;
	} else if (a <= -3e-90) {
		tmp = t * (y / (a - z));
	} else if (a <= 5.2e+20) {
		tmp = fma(a, (t / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(z / a), x)
	tmp = 0.0
	if (a <= -7e+93)
		tmp = t_1;
	elseif (a <= -3e-90)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (a <= 5.2e+20)
		tmp = fma(a, Float64(t / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7e+93], t$95$1, If[LessEqual[a, -3e-90], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.2e+20], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.99999999999999996e93 or 5.2e20 < a

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 60.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 59.9%

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

   if -6.99999999999999996e93 < a < -3.0000000000000002e-90

   1. Initial program 76.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 35.9%

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

   if -3.0000000000000002e-90 < a < 5.2e20

   1. Initial program 60.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 49.2%

       \[\leadsto \mathsf{fma}\left(a, \frac{t}{\color{blue}{z}}, t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+33} \lor \neg \left(y \leq 1.95 \cdot 10^{-46}\right):\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -1.8e+33) (not (<= y 1.95e-46)))
   (* (- t x) (/ y (- a z)))
   (fma (- x t) (/ z (- a z)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.8e+33) || !(y <= 1.95e-46)) {
		tmp = (t - x) * (y / (a - z));
	} else {
		tmp = fma((x - t), (z / (a - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -1.8e+33) || !(y <= 1.95e-46))
		tmp = Float64(Float64(t - x) * Float64(y / Float64(a - z)));
	else
		tmp = fma(Float64(x - t), Float64(z / Float64(a - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -1.8e+33], N[Not[LessEqual[y, 1.95e-46]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8000000000000001e33 or 1.9500000000000001e-46 < y

   1. Initial program 68.2%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -1.8000000000000001e33 < y < 1.9500000000000001e-46

   1. Initial program 66.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.4%

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{z}}{a - z}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+33} \lor \neg \left(y \leq 1.95 \cdot 10^{-46}\right):\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+41} \lor \neg \left(z \leq 2 \cdot 10^{+138}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.8e+41) (not (<= z 2e+138)))
   (* (- t) (/ z (- a z)))
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.8e+41) || !(z <= 2e+138)) {
		tmp = -t * (z / (a - z));
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.8e+41) || !(z <= 2e+138))
		tmp = Float64(Float64(-t) * Float64(z / Float64(a - z)));
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.8e+41], N[Not[LessEqual[z, 2e+138]], $MachinePrecision]], N[((-t) * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.7999999999999994e41 or 2.0000000000000001e138 < z

   1. Initial program 39.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.7%

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

   if -7.7999999999999994e41 < z < 2.0000000000000001e138

   1. Initial program 84.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 64.2

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

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+41} \lor \neg \left(z \leq 2 \cdot 10^{+138}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -30500000000 \lor \neg \left(z \leq 2.3 \cdot 10^{+138}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -30500000000.0) (not (<= z 2.3e+138)))
   (* (- t) (/ (- y z) z))
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -30500000000.0) || !(z <= 2.3e+138)) {
		tmp = -t * ((y - z) / z);
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -30500000000.0) || !(z <= 2.3e+138))
		tmp = Float64(Float64(-t) * Float64(Float64(y - z) / z));
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -30500000000.0], N[Not[LessEqual[z, 2.3e+138]], $MachinePrecision]], N[((-t) * N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.05e10 or 2.30000000000000008e138 < z

   1. Initial program 41.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.4%

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

   if -3.05e10 < z < 2.30000000000000008e138

   1. Initial program 84.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.0

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

   5. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -30500000000 \lor \neg \left(z \leq 2.3 \cdot 10^{+138}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{y - z}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+42} \lor \neg \left(z \leq 2.3 \cdot 10^{+138}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.38e+42) (not (<= z 2.3e+138)))
   (* (- t) -1.0)
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.38e+42) || !(z <= 2.3e+138)) {
		tmp = -t * -1.0;
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.38e+42) || !(z <= 2.3e+138))
		tmp = Float64(Float64(-t) * -1.0);
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.38e+42], N[Not[LessEqual[z, 2.3e+138]], $MachinePrecision]], N[((-t) * -1.0), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3800000000000001e42 or 2.30000000000000008e138 < z

   1. Initial program 39.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 64.2%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -1.3800000000000001e42 < z < 2.30000000000000008e138

   1. Initial program 84.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 64.2

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

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{+42} \lor \neg \left(z \leq 2.3 \cdot 10^{+138}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-87}:\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+132}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.8e-87)
   (* (- t) -1.0)
   (if (<= z 2.9e+132) (* t (/ y (- a z))) (fma a (/ t z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.8e-87) {
		tmp = -t * -1.0;
	} else if (z <= 2.9e+132) {
		tmp = t * (y / (a - z));
	} else {
		tmp = fma(a, (t / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.8e-87)
		tmp = Float64(Float64(-t) * -1.0);
	elseif (z <= 2.9e+132)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	else
		tmp = fma(a, Float64(t / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.8e-87], N[((-t) * -1.0), $MachinePrecision], If[LessEqual[z, 2.9e+132], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.7999999999999997e-87

   1. Initial program 49.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 48.3%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -6.7999999999999997e-87 < z < 2.8999999999999999e132

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 38.7

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

   5. Applied rewrites 38.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 31.4%

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

   if 2.8999999999999999e132 < z

   1. Initial program 40.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 69.0%

       \[\leadsto \mathsf{fma}\left(a, \frac{t}{\color{blue}{z}}, t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 33.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-205} \lor \neg \left(z \leq 1.6 \cdot 10^{-13}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.4e-205) (not (<= z 1.6e-13))) (* (- t) -1.0) (/ (* t y) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.4e-205) || !(z <= 1.6e-13)) {
		tmp = -t * -1.0;
	} else {
		tmp = (t * y) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 ((z <= (-3.4d-205)) .or. (.not. (z <= 1.6d-13))) then
        tmp = -t * (-1.0d0)
    else
        tmp = (t * y) / 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 ((z <= -3.4e-205) || !(z <= 1.6e-13)) {
		tmp = -t * -1.0;
	} else {
		tmp = (t * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.4e-205) or not (z <= 1.6e-13):
		tmp = -t * -1.0
	else:
		tmp = (t * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.4e-205) || !(z <= 1.6e-13))
		tmp = Float64(Float64(-t) * -1.0);
	else
		tmp = Float64(Float64(t * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.4e-205) || ~((z <= 1.6e-13)))
		tmp = -t * -1.0;
	else
		tmp = (t * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.4e-205], N[Not[LessEqual[z, 1.6e-13]], $MachinePrecision]], N[((-t) * -1.0), $MachinePrecision], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4000000000000002e-205 or 1.6e-13 < z

   1. Initial program 54.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 52.5

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 42.6%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -3.4000000000000002e-205 < z < 1.6e-13

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 30.2%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-205} \lor \neg \left(z \leq 1.6 \cdot 10^{-13}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-198}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-198)
   (fma a (/ t z) t)
   (if (<= z 1.6e-13) (/ (* t y) a) (* (- t) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-198) {
		tmp = fma(a, (t / z), t);
	} else if (z <= 1.6e-13) {
		tmp = (t * y) / a;
	} else {
		tmp = -t * -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-198)
		tmp = fma(a, Float64(t / z), t);
	elseif (z <= 1.6e-13)
		tmp = Float64(Float64(t * y) / a);
	else
		tmp = Float64(Float64(-t) * -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-198], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.6e-13], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], N[((-t) * -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e-198

   1. Initial program 58.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 41.1%

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

   if -2.2e-198 < z < 1.6e-13

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.6%

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

   if 1.6e-13 < z

   1. Initial program 48.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 46.5%

       \[\leadsto \left(-t\right) \cdot -1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-t\right) \cdot -1 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* (- t) -1.0))

C

double code(double x, double y, double z, double t, double a) {
	return -t * -1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 = -t * (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -t * -1.0;
}

Python

def code(x, y, z, t, a):
	return -t * -1.0

Julia

function code(x, y, z, t, a)
	return Float64(Float64(-t) * -1.0)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -t * -1.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[((-t) * -1.0), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 49.1

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

5. Applied rewrites 49.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 35.3%

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

9. Taylor expanded in z around inf

   \[\leadsto \left(-t\right) \cdot -1 \]
10. Step-by-step derivation

11. Applied rewrites 31.2%

    \[\leadsto \left(-t\right) \cdot -1 \]
12. Add Preprocessing

Alternative 17: 20.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x + \left(t - x\right) \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 + (t - x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 22.9

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

5. Applied rewrites 22.9%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
6. Add Preprocessing

Alternative 18: 2.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x + \left(-x\right) \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 + -x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + (-x)), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 22.9

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

5. Applied rewrites 22.9%

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

6. Taylor expanded in x around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 2.8%

   \[\leadsto x + \left(-x\right) \]
9. Add Preprocessing

Developer Target 1: 84.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    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 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - 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 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024359 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

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