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

Percentage Accurate: 68.3% → 87.3%

Time: 19.3s

Alternatives: 20

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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 - x) * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 - x) * (z - t)) / (a - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-15} \lor \neg \left(y \leq 9.5 \cdot 10^{-97}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{\left(z - t\right) \cdot y}{x}}{a - t}, -1, \frac{z}{a - t}\right) - \left(\frac{t}{a - t} - -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -5.8e-15) (not (<= y 9.5e-97)))
   (fma (- y x) (/ (- z t) (- a t)) x)
   (*
    (- x)
    (-
     (fma (/ (/ (* (- z t) y) x) (- a t)) -1.0 (/ z (- a t)))
     (- (/ t (- a t)) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -5.8e-15) || !(y <= 9.5e-97)) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = -x * (fma(((((z - t) * y) / x) / (a - t)), -1.0, (z / (a - t))) - ((t / (a - t)) - -1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -5.8e-15) || !(y <= 9.5e-97))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = Float64(Float64(-x) * Float64(fma(Float64(Float64(Float64(Float64(z - t) * y) / x) / Float64(a - t)), -1.0, Float64(z / Float64(a - t))) - Float64(Float64(t / Float64(a - t)) - -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -5.8e-15], N[Not[LessEqual[y, 9.5e-97]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[((-x) * N[(N[(N[(N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000037e-15 or 9.5000000000000001e-97 < y

   1. Initial program 65.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 91.9

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

   4. Applied rewrites 91.9%

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

   if -5.80000000000000037e-15 < y < 9.5000000000000001e-97

   1. Initial program 68.8%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{\left(z - t\right) \cdot y}{x}}{a - t}, -1, \frac{z}{a - t}\right) - \left(\frac{t}{a - t} + 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-15} \lor \neg \left(y \leq 9.5 \cdot 10^{-97}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{\left(z - t\right) \cdot y}{x}}{a - t}, -1, \frac{z}{a - t}\right) - \left(\frac{t}{a - t} - -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-275} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(y - x\right) \cdot z}{t}, -1, y\right) + \frac{\left(y - x\right) \cdot a}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (or (<= t_1 -2e-275) (not (<= t_1 0.0)))
     (fma (- y x) (/ (- z t) (- a t)) x)
     (+ (fma (/ (* (- y x) z) t) -1.0 y) (/ (* (- y x) a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if ((t_1 <= -2e-275) || !(t_1 <= 0.0)) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = fma((((y - x) * z) / t), -1.0, y) + (((y - x) * a) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if ((t_1 <= -2e-275) || !(t_1 <= 0.0))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = Float64(fma(Float64(Float64(Float64(y - x) * z) / t), -1.0, y) + Float64(Float64(Float64(y - x) * a) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-275], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] * -1.0 + y), $MachinePrecision] + N[(N[(N[(y - x), $MachinePrecision] * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.99999999999999987e-275 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 71.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if -1.99999999999999987e-275 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 5.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 99.8

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

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(y - x\right) \cdot z}{t}, -1, y\right) - \left(-\frac{\left(y - x\right) \cdot a}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2 \cdot 10^{-275} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(y - x\right) \cdot z}{t}, -1, y\right) + \frac{\left(y - x\right) \cdot a}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-275} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}, -1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (or (<= t_1 -2e-275) (not (<= t_1 0.0)))
     (fma (- y x) (/ (- z t) (- a t)) x)
     (fma (/ (* (- y x) (- z a)) t) -1.0 y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if ((t_1 <= -2e-275) || !(t_1 <= 0.0)) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = fma((((y - x) * (z - a)) / t), -1.0, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if ((t_1 <= -2e-275) || !(t_1 <= 0.0))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = fma(Float64(Float64(Float64(y - x) * Float64(z - a)) / t), -1.0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-275], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(y - x), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -1.0 + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.99999999999999987e-275 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 71.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if -1.99999999999999987e-275 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 5.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}, -1, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2 \cdot 10^{-275} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(y - x\right) \cdot \left(z - a\right)}{t}, -1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-275} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (or (<= t_1 -2e-275) (not (<= t_1 0.0)))
     (fma (- y x) (/ (- z t) (- a t)) x)
     (fma (/ (- y x) t) a y))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if ((t_1 <= -2e-275) || !(t_1 <= 0.0)) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else {
		tmp = fma(((y - x) / t), a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if ((t_1 <= -2e-275) || !(t_1 <= 0.0))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	else
		tmp = fma(Float64(Float64(y - x) / t), a, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-275], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * a + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -1.99999999999999987e-275 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 71.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if -1.99999999999999987e-275 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 5.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 65.9

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

   8. Applied rewrites 65.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2 \cdot 10^{-275} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-130}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y - x}{t}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-219}:\\ \;\;\;\;y\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{-t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ z a) x)))
   (if (<= a -7e-54)
     t_1
     (if (<= a -1.35e-130)
       (* (- z) (/ (- y x) t))
       (if (<= a -4.8e-219)
         y
         (if (<= a 1.3e-114) (/ (* (- y x) z) (- t)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), (z / a), x);
	double tmp;
	if (a <= -7e-54) {
		tmp = t_1;
	} else if (a <= -1.35e-130) {
		tmp = -z * ((y - x) / t);
	} else if (a <= -4.8e-219) {
		tmp = y;
	} else if (a <= 1.3e-114) {
		tmp = ((y - x) * z) / -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(z / a), x)
	tmp = 0.0
	if (a <= -7e-54)
		tmp = t_1;
	elseif (a <= -1.35e-130)
		tmp = Float64(Float64(-z) * Float64(Float64(y - x) / t));
	elseif (a <= -4.8e-219)
		tmp = y;
	elseif (a <= 1.3e-114)
		tmp = Float64(Float64(Float64(y - x) * z) / Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7e-54], t$95$1, If[LessEqual[a, -1.35e-130], N[((-z) * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.8e-219], y, If[LessEqual[a, 1.3e-114], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / (-t)), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.99999999999999964e-54 or 1.30000000000000007e-114 < a

   1. Initial program 72.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 66.2

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

   7. Applied rewrites 66.2%

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

   if -6.99999999999999964e-54 < a < -1.34999999999999996e-130

   1. Initial program 58.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 16.1

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in t around inf

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

       1. associate-/l* N/A

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

       2. sub-div N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. sub-div N/A

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

       8. lower-/.f64 N/A

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

       9. lift--.f64 63.4

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

   11. Applied rewrites 63.4%

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

   if -1.34999999999999996e-130 < a < -4.80000000000000028e-219

   1. Initial program 46.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 63.5%

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

   if -4.80000000000000028e-219 < a < 1.30000000000000007e-114

   1. Initial program 63.4%

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

   3. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.0

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

   8. Applied rewrites 53.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-130}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y - x}{t}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-219}:\\ \;\;\;\;y\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{-t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-z\right) \cdot \frac{y - x}{t}\\ t_2 := \mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-219}:\\ \;\;\;\;y\\ \mathbf{elif}\;a \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z) (/ (- y x) t))) (t_2 (fma (- y x) (/ z a) x)))
   (if (<= a -7e-54)
     t_2
     (if (<= a -1.35e-130)
       t_1
       (if (<= a -1.45e-219) y (if (<= a 1.04e-109) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -z * ((y - x) / t);
	double t_2 = fma((y - x), (z / a), x);
	double tmp;
	if (a <= -7e-54) {
		tmp = t_2;
	} else if (a <= -1.35e-130) {
		tmp = t_1;
	} else if (a <= -1.45e-219) {
		tmp = y;
	} else if (a <= 1.04e-109) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-z) * Float64(Float64(y - x) / t))
	t_2 = fma(Float64(y - x), Float64(z / a), x)
	tmp = 0.0
	if (a <= -7e-54)
		tmp = t_2;
	elseif (a <= -1.35e-130)
		tmp = t_1;
	elseif (a <= -1.45e-219)
		tmp = y;
	elseif (a <= 1.04e-109)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-z) * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7e-54], t$95$2, If[LessEqual[a, -1.35e-130], t$95$1, If[LessEqual[a, -1.45e-219], y, If[LessEqual[a, 1.04e-109], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.99999999999999964e-54 or 1.03999999999999996e-109 < a

   1. Initial program 72.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 66.2

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

   7. Applied rewrites 66.2%

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

   if -6.99999999999999964e-54 < a < -1.34999999999999996e-130 or -1.44999999999999992e-219 < a < 1.03999999999999996e-109

   1. Initial program 62.4%

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

   3. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 16.6

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

   8. Applied rewrites 16.6%

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

   9. Taylor expanded in t around inf

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

       1. associate-/l* N/A

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

       2. sub-div N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. sub-div N/A

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

       8. lower-/.f64 N/A

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

       9. lift--.f64 52.7

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

   11. Applied rewrites 52.7%

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

   if -1.34999999999999996e-130 < a < -1.44999999999999992e-219

   1. Initial program 46.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 63.5%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-130}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y - x}{t}\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-219}:\\ \;\;\;\;y\\ \mathbf{elif}\;a \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+177} \lor \neg \left(z \leq 1.5 \cdot 10^{+107}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -8e+177) (not (<= z 1.5e+107)))
   (* (- y x) (/ z (- a t)))
   (fma y (/ (- z t) (- a t)) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -8e+177) || !(z <= 1.5e+107)) {
		tmp = (y - x) * (z / (a - t));
	} else {
		tmp = fma(y, ((z - t) / (a - t)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -8e+177) || !(z <= 1.5e+107))
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	else
		tmp = fma(y, Float64(Float64(z - t) / Float64(a - t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -8e+177], N[Not[LessEqual[z, 1.5e+107]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.0000000000000001e177 or 1.50000000000000012e107 < z

   1. Initial program 65.8%

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

   3. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 62.9

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

   5. Applied rewrites 62.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 85.9

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

   7. Applied rewrites 85.9%

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

   if -8.0000000000000001e177 < z < 1.50000000000000012e107

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 72.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+177} \lor \neg \left(z \leq 1.5 \cdot 10^{+107}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-28} \lor \neg \left(a \leq 6.8 \cdot 10^{-19}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -9e-28) (not (<= a 6.8e-19)))
   (fma (- y x) (/ (- z t) a) x)
   (* y (/ (- z t) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -9e-28) || !(a <= 6.8e-19)) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = y * ((z - t) / (a - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -9e-28) || !(a <= 6.8e-19))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -9e-28], N[Not[LessEqual[a, 6.8e-19]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.9999999999999996e-28 or 6.8000000000000004e-19 < a

   1. Initial program 73.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -8.9999999999999996e-28 < a < 6.8000000000000004e-19

   1. Initial program 60.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 65.2

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

   7. Applied rewrites 65.2%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 66.4

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

   10. Applied rewrites 66.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-28} \lor \neg \left(a \leq 6.8 \cdot 10^{-19}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -1.02 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-65}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -1.02e-25)
     t_1
     (if (<= a -6.8e-129) (* (/ (- z a) t) x) (if (<= a 2.4e-65) y t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -1.02e-25) {
		tmp = t_1;
	} else if (a <= -6.8e-129) {
		tmp = ((z - a) / t) * x;
	} else if (a <= 2.4e-65) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -1.02e-25)
		tmp = t_1;
	elseif (a <= -6.8e-129)
		tmp = Float64(Float64(Float64(z - a) / t) * x);
	elseif (a <= 2.4e-65)
		tmp = y;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.02e-25], t$95$1, If[LessEqual[a, -6.8e-129], N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 2.4e-65], y, t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999998e-25 or 2.4000000000000002e-65 < a

   1. Initial program 72.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.9%

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

   if -1.01999999999999998e-25 < a < -6.80000000000000026e-129

   1. Initial program 60.1%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 53.5

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

   8. Applied rewrites 53.5%

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

   if -6.80000000000000026e-129 < a < 2.4000000000000002e-65

   1. Initial program 61.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.2%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{z - a}{t} \cdot x\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-65}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+110} \lor \neg \left(z \leq 8 \cdot 10^{-19}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.3e+110) (not (<= z 8e-19)))
   (* (- y x) (/ z (- a t)))
   (* y (/ (- z t) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.3e+110) || !(z <= 8e-19)) {
		tmp = (y - x) * (z / (a - t));
	} else {
		tmp = y * ((z - t) / (a - t));
	}
	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 <= (-1.3d+110)) .or. (.not. (z <= 8d-19))) then
        tmp = (y - x) * (z / (a - t))
    else
        tmp = y * ((z - t) / (a - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.3e+110) || !(z <= 8e-19)) {
		tmp = (y - x) * (z / (a - t));
	} else {
		tmp = y * ((z - t) / (a - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.3e+110) or not (z <= 8e-19):
		tmp = (y - x) * (z / (a - t))
	else:
		tmp = y * ((z - t) / (a - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.3e+110) || !(z <= 8e-19))
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	else
		tmp = Float64(y * Float64(Float64(z - t) / Float64(a - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.3e+110) || ~((z <= 8e-19)))
		tmp = (y - x) * (z / (a - t));
	else
		tmp = y * ((z - t) / (a - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.3e+110], N[Not[LessEqual[z, 8e-19]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e110 or 7.9999999999999998e-19 < z

   1. Initial program 68.2%

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

   3. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 60.4

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

   5. Applied rewrites 60.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if -1.3e110 < z < 7.9999999999999998e-19

   1. Initial program 65.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.9

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 61.0

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

   10. Applied rewrites 61.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+110} \lor \neg \left(z \leq 8 \cdot 10^{-19}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-25} \lor \neg \left(z \leq 3.5 \cdot 10^{-79}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-t}{a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.3e-25) (not (<= z 3.5e-79)))
   (* (- y x) (/ z (- a t)))
   (* y (/ (- t) (- a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.3e-25) || !(z <= 3.5e-79)) {
		tmp = (y - x) * (z / (a - t));
	} else {
		tmp = y * (-t / (a - t));
	}
	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 <= (-4.3d-25)) .or. (.not. (z <= 3.5d-79))) then
        tmp = (y - x) * (z / (a - t))
    else
        tmp = y * (-t / (a - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.3e-25) || !(z <= 3.5e-79)) {
		tmp = (y - x) * (z / (a - t));
	} else {
		tmp = y * (-t / (a - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.3e-25) or not (z <= 3.5e-79):
		tmp = (y - x) * (z / (a - t))
	else:
		tmp = y * (-t / (a - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.3e-25) || !(z <= 3.5e-79))
		tmp = Float64(Float64(y - x) * Float64(z / Float64(a - t)));
	else
		tmp = Float64(y * Float64(Float64(-t) / Float64(a - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.3e-25) || ~((z <= 3.5e-79)))
		tmp = (y - x) * (z / (a - t));
	else
		tmp = y * (-t / (a - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.3e-25], N[Not[LessEqual[z, 3.5e-79]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[((-t) / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.29999999999999976e-25 or 3.5000000000000003e-79 < z

   1. Initial program 67.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-div N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 57.4

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

   5. Applied rewrites 57.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 71.7

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

   7. Applied rewrites 71.7%

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

   if -4.29999999999999976e-25 < z < 3.5000000000000003e-79

   1. Initial program 65.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 72.4

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 60.9

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

   10. Applied rewrites 60.9%

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

   11. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

          \[\leadsto y \cdot \frac{\mathsf{neg}\left(t\right)}{a - t} \]

       2. lower-neg.f64 56.1

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

   13. Applied rewrites 56.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-25} \lor \neg \left(z \leq 3.5 \cdot 10^{-79}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-29} \lor \neg \left(a \leq 6.5 \cdot 10^{-74}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.25e-29) (not (<= a 6.5e-74)))
   (fma (- y x) (/ z a) x)
   (* y (/ (- z t) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.25e-29) || !(a <= 6.5e-74)) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = y * ((z - t) / -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.25e-29) || !(a <= 6.5e-74))
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = Float64(y * Float64(Float64(z - t) / Float64(-t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.25e-29], N[Not[LessEqual[a, 6.5e-74]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2499999999999999e-29 or 6.5000000000000002e-74 < a

   1. Initial program 72.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if -2.2499999999999999e-29 < a < 6.5000000000000002e-74

   1. Initial program 59.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 73.5

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

   4. Applied rewrites 73.5%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 63.5

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 66.5

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

   10. Applied rewrites 66.5%

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

   11. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 58.8

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

   13. Applied rewrites 58.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-29} \lor \neg \left(a \leq 6.5 \cdot 10^{-74}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+117} \lor \neg \left(t \leq 4.5 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot \frac{-t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.4e+117) (not (<= t 4.5e+55)))
   (* y (/ (- t) (- a t)))
   (fma (- y x) (/ z a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.4e+117) || !(t <= 4.5e+55)) {
		tmp = y * (-t / (a - t));
	} else {
		tmp = fma((y - x), (z / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.4e+117) || !(t <= 4.5e+55))
		tmp = Float64(y * Float64(Float64(-t) / Float64(a - t)));
	else
		tmp = fma(Float64(y - x), Float64(z / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.4e+117], N[Not[LessEqual[t, 4.5e+55]], $MachinePrecision]], N[(y * N[((-t) / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.40000000000000028e117 or 4.49999999999999998e55 < t

   1. Initial program 39.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift--.f64 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 63.8

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

   10. Applied rewrites 63.8%

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

   11. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

          \[\leadsto y \cdot \frac{\mathsf{neg}\left(t\right)}{a - t} \]

       2. lower-neg.f64 55.1

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

   13. Applied rewrites 55.1%

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

   if -4.40000000000000028e117 < t < 4.49999999999999998e55

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 64.3

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

   7. Applied rewrites 64.3%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+117} \lor \neg \left(t \leq 4.5 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot \frac{-t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{-12} \lor \neg \left(a \leq 2.7 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.55e-12) (not (<= a 2.7e-65)))
   (fma z (/ y a) x)
   (fma (/ (- y x) t) a y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.55e-12) || !(a <= 2.7e-65)) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = fma(((y - x) / t), a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.55e-12) || !(a <= 2.7e-65))
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = fma(Float64(Float64(y - x) / t), a, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.55e-12], N[Not[LessEqual[a, 2.7e-65]], $MachinePrecision]], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * a + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.54999999999999984e-12 or 2.6999999999999999e-65 < a

   1. Initial program 71.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.3%

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

   if -2.54999999999999984e-12 < a < 2.6999999999999999e-65

   1. Initial program 61.2%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 42.0

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

   8. Applied rewrites 42.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{-12} \lor \neg \left(a \leq 2.7 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.1e+136)
   y
   (if (<= t 3.6e+120) (fma (- y x) (/ z a) x) (fma (/ (- y x) t) a y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1e+136) {
		tmp = y;
	} else if (t <= 3.6e+120) {
		tmp = fma((y - x), (z / a), x);
	} else {
		tmp = fma(((y - x) / t), a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.1e+136)
		tmp = y;
	elseif (t <= 3.6e+120)
		tmp = fma(Float64(y - x), Float64(z / a), x);
	else
		tmp = fma(Float64(Float64(y - x) / t), a, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.1e+136], y, If[LessEqual[t, 3.6e+120], N[(N[(y - x), $MachinePrecision] * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * a + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1e136

   1. Initial program 24.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.8%

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

   if -1.1e136 < t < 3.60000000000000016e120

   1. Initial program 81.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 59.5

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

   7. Applied rewrites 59.5%

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

   if 3.60000000000000016e120 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 47.2

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

   8. Applied rewrites 47.2%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.1e+136)
   y
   (if (<= t 2.8e+120) (fma z (/ (- y x) a) x) (fma (/ (- y x) t) a y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.1e+136) {
		tmp = y;
	} else if (t <= 2.8e+120) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = fma(((y - x) / t), a, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.1e+136)
		tmp = y;
	elseif (t <= 2.8e+120)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = fma(Float64(Float64(y - x) / t), a, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.1e+136], y, If[LessEqual[t, 2.8e+120], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * a + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1e136

   1. Initial program 24.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.8%

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

   if -1.1e136 < t < 2.8000000000000001e120

   1. Initial program 81.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 2.8000000000000001e120 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-div N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 47.2

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

   8. Applied rewrites 47.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+67}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.8e+67) y (if (<= t 4.7e+55) (fma z (/ y a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.8e+67) {
		tmp = y;
	} else if (t <= 4.7e+55) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.8e+67)
		tmp = y;
	elseif (t <= 4.7e+55)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.8e+67], y, If[LessEqual[t, 4.7e+55], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7999999999999999e67 or 4.7000000000000001e55 < t

   1. Initial program 40.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 46.6%

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

   if -1.7999999999999999e67 < t < 4.7000000000000001e55

   1. Initial program 86.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.0%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+67}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+98}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1700000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-9}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5e+98) x (if (<= a -1700000.0) (+ x y) (if (<= a 8e-9) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5e+98) {
		tmp = x;
	} else if (a <= -1700000.0) {
		tmp = x + y;
	} else if (a <= 8e-9) {
		tmp = y;
	} else {
		tmp = x;
	}
	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 (a <= (-5d+98)) then
        tmp = x
    else if (a <= (-1700000.0d0)) then
        tmp = x + y
    else if (a <= 8d-9) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5e+98) {
		tmp = x;
	} else if (a <= -1700000.0) {
		tmp = x + y;
	} else if (a <= 8e-9) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -5e+98:
		tmp = x
	elif a <= -1700000.0:
		tmp = x + y
	elif a <= 8e-9:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5e+98)
		tmp = x;
	elseif (a <= -1700000.0)
		tmp = Float64(x + y);
	elseif (a <= 8e-9)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -5e+98)
		tmp = x;
	elseif (a <= -1700000.0)
		tmp = x + y;
	elseif (a <= 8e-9)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5e+98], x, If[LessEqual[a, -1700000.0], N[(x + y), $MachinePrecision], If[LessEqual[a, 8e-9], y, x]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.9999999999999998e98 or 8.0000000000000005e-9 < a

   1. Initial program 68.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 45.7%

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

   if -4.9999999999999998e98 < a < -1.7e6

   1. Initial program 91.9%

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

   3. Taylor expanded in t around inf

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

      1. lift--.f64 23.3

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto x + y \]
   7. Step-by-step derivation

   8. Applied rewrites 47.6%

      \[\leadsto x + y \]

   if -1.7e6 < a < 8.0000000000000005e-9

   1. Initial program 61.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.6%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+98}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1700000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-9}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 39.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -24000000:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-9}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -24000000.0) x (if (<= a 8e-9) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -24000000.0) {
		tmp = x;
	} else if (a <= 8e-9) {
		tmp = y;
	} else {
		tmp = x;
	}
	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 (a <= (-24000000.0d0)) then
        tmp = x
    else if (a <= 8d-9) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -24000000.0) {
		tmp = x;
	} else if (a <= 8e-9) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -24000000.0:
		tmp = x
	elif a <= 8e-9:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -24000000.0)
		tmp = x;
	elseif (a <= 8e-9)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -24000000.0)
		tmp = x;
	elseif (a <= 8e-9)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -24000000.0], x, If[LessEqual[a, 8e-9], y, x]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4e7 or 8.0000000000000005e-9 < a

   1. Initial program 72.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.1%

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

   if -2.4e7 < a < 8.0000000000000005e-9

   1. Initial program 61.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.6%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -24000000:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-9}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 25.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Applied rewrites 22.5%

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

6. Final simplification 22.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 87.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025043 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

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