Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Percentage Accurate: 85.3% → 96.7%

Time: 6.0s

Alternatives: 12

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) t) (- a z)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+305)))
     (* (- y z) (/ t (- a z)))
     t_1)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * t) / (a - z));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+305)) {
		tmp = (y - z) * (t / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * t) / (a - z));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e+305)) {
		tmp = (y - z) * (t / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) * t) / (a - z))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e+305):
		tmp = (y - z) * (t / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+305))
		tmp = Float64(Float64(y - z) * Float64(t / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) * t) / (a - z));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e+305)))
		tmp = (y - z) * (t / (a - z));
	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 - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+305]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))) < -inf.0 or 5.00000000000000009e305 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)))

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))) < 5.00000000000000009e305

   1. Initial program 99.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty \lor \neg \left(x + \frac{\left(y - z\right) \cdot t}{a - z} \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-91}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.06e-101)
   (fma (/ (- z y) z) t x)
   (if (<= z 2.7e-91)
     (+ x (/ (* (- y z) t) a))
     (if (<= z 6.1e+88) (fma (- z y) (/ t z) x) (fma (/ z (- a z)) (- t) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.06e-101) {
		tmp = fma(((z - y) / z), t, x);
	} else if (z <= 2.7e-91) {
		tmp = x + (((y - z) * t) / a);
	} else if (z <= 6.1e+88) {
		tmp = fma((z - y), (t / z), x);
	} else {
		tmp = fma((z / (a - z)), -t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.06e-101)
		tmp = fma(Float64(Float64(z - y) / z), t, x);
	elseif (z <= 2.7e-91)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / a));
	elseif (z <= 6.1e+88)
		tmp = fma(Float64(z - y), Float64(t / z), x);
	else
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.06e-101], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 2.7e-91], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.1e+88], N[(N[(z - y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.0600000000000001e-101

   1. Initial program 78.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -1.0600000000000001e-101 < z < 2.6999999999999997e-91

   1. Initial program 97.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if 2.6999999999999997e-91 < z < 6.0999999999999998e88

   1. Initial program 91.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 76.0

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

   5. Applied rewrites 76.0%

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

   7. Applied rewrites 80.8%

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

   if 6.0999999999999998e88 < z

   1. Initial program 60.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

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

Alternative 3: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-8}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e-103)
   (fma (/ (- z y) z) t x)
   (if (<= z 1.25e-55)
     (fma (/ y a) t x)
     (if (<= z 6.8e-8)
       (* (- y z) (/ t (- a z)))
       (fma (/ z (- a z)) (- t) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e-103) {
		tmp = fma(((z - y) / z), t, x);
	} else if (z <= 1.25e-55) {
		tmp = fma((y / a), t, x);
	} else if (z <= 6.8e-8) {
		tmp = (y - z) * (t / (a - z));
	} else {
		tmp = fma((z / (a - z)), -t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e-103)
		tmp = fma(Float64(Float64(z - y) / z), t, x);
	elseif (z <= 1.25e-55)
		tmp = fma(Float64(y / a), t, x);
	elseif (z <= 6.8e-8)
		tmp = Float64(Float64(y - z) * Float64(t / Float64(a - z)));
	else
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e-103], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 1.25e-55], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 6.8e-8], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.8000000000000001e-103

   1. Initial program 78.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -3.8000000000000001e-103 < z < 1.25e-55

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if 1.25e-55 < z < 6.8e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if 6.8e-8 < z

   1. Initial program 69.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 84.4

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

   5. Applied rewrites 84.4%

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

Alternative 4: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+38}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(z - y\right) \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e+38)
   (+ t x)
   (if (<= z 1.1e-48)
     (fma (/ y a) t x)
     (if (<= z 1.1e-12) (/ (* (- z y) t) z) (+ t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e+38) {
		tmp = t + x;
	} else if (z <= 1.1e-48) {
		tmp = fma((y / a), t, x);
	} else if (z <= 1.1e-12) {
		tmp = ((z - y) * t) / z;
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e+38)
		tmp = Float64(t + x);
	elseif (z <= 1.1e-48)
		tmp = fma(Float64(y / a), t, x);
	elseif (z <= 1.1e-12)
		tmp = Float64(Float64(Float64(z - y) * t) / z);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e+38], N[(t + x), $MachinePrecision], If[LessEqual[z, 1.1e-48], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 1.1e-12], N[(N[(N[(z - y), $MachinePrecision] * t), $MachinePrecision] / z), $MachinePrecision], N[(t + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.50000000000000002e38 or 1.09999999999999996e-12 < z

   1. Initial program 70.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -3.50000000000000002e38 < z < 1.10000000000000006e-48

   1. Initial program 96.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 1.10000000000000006e-48 < z < 1.09999999999999996e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.1%

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

Alternative 5: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+71}:\\ \;\;\;\;x + \frac{t \cdot y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.55e+39)
   (fma (/ (- z y) z) t x)
   (if (<= z 4.6e+71) (+ x (/ (* t y) (- a z))) (fma (/ z (- a z)) (- t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.55e+39) {
		tmp = fma(((z - y) / z), t, x);
	} else if (z <= 4.6e+71) {
		tmp = x + ((t * y) / (a - z));
	} else {
		tmp = fma((z / (a - z)), -t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.55e+39)
		tmp = fma(Float64(Float64(z - y) / z), t, x);
	elseif (z <= 4.6e+71)
		tmp = Float64(x + Float64(Float64(t * y) / Float64(a - z)));
	else
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.55e+39], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 4.6e+71], N[(x + N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5499999999999999e39

   1. Initial program 71.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.5499999999999999e39 < z < 4.6000000000000005e71

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 4.6000000000000005e71 < z

   1. Initial program 62.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 89.9

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

   5. Applied rewrites 89.9%

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

Alternative 6: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-127} \lor \neg \left(z \leq 1.65 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3e-127) (not (<= z 1.65e-62)))
   (fma (- z y) (/ t z) x)
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3e-127) || !(z <= 1.65e-62)) {
		tmp = fma((z - y), (t / z), x);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3e-127) || !(z <= 1.65e-62))
		tmp = fma(Float64(z - y), Float64(t / z), x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3e-127], N[Not[LessEqual[z, 1.65e-62]], $MachinePrecision]], N[(N[(z - y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.00000000000000009e-127 or 1.65000000000000002e-62 < z

   1. Initial program 77.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 79.0%

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

   if -3.00000000000000009e-127 < z < 1.65000000000000002e-62

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-127} \lor \neg \left(z \leq 1.65 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - y, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e-103)
   (fma (/ (- z y) z) t x)
   (if (<= z 1.65e-62) (fma (/ y a) t x) (fma (- z y) (/ t z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e-103) {
		tmp = fma(((z - y) / z), t, x);
	} else if (z <= 1.65e-62) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = fma((z - y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e-103)
		tmp = fma(Float64(Float64(z - y) / z), t, x);
	elseif (z <= 1.65e-62)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = fma(Float64(z - y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e-103], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 1.65e-62], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(z - y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8000000000000001e-103

   1. Initial program 78.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -3.8000000000000001e-103 < z < 1.65000000000000002e-62

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if 1.65000000000000002e-62 < z

   1. Initial program 75.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. mul-1-neg N/A

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

      18. fp-cancel-sign-sub N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f64 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 80.8%

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

Alternative 8: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+38} \lor \neg \left(z \leq 3.9 \cdot 10^{-28}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.5e+38) (not (<= z 3.9e-28))) (+ t x) (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.5e+38) || !(z <= 3.9e-28)) {
		tmp = t + x;
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.5e+38) || !(z <= 3.9e-28))
		tmp = Float64(t + x);
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.5e+38], N[Not[LessEqual[z, 3.9e-28]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000002e38 or 3.89999999999999999e-28 < z

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if -3.50000000000000002e38 < z < 3.89999999999999999e-28

   1. Initial program 97.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+38} \lor \neg \left(z \leq 3.9 \cdot 10^{-28}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+110} \lor \neg \left(y \leq 2.15 \cdot 10^{+171}\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -1e+110) (not (<= y 2.15e+171))) (* y (/ t a)) (+ t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1e+110) || !(y <= 2.15e+171)) {
		tmp = y * (t / a);
	} else {
		tmp = t + 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 ((y <= (-1d+110)) .or. (.not. (y <= 2.15d+171))) then
        tmp = y * (t / a)
    else
        tmp = t + 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 ((y <= -1e+110) || !(y <= 2.15e+171)) {
		tmp = y * (t / a);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -1e+110) or not (y <= 2.15e+171):
		tmp = y * (t / a)
	else:
		tmp = t + x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -1e+110) || !(y <= 2.15e+171))
		tmp = Float64(y * Float64(t / a));
	else
		tmp = Float64(t + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -1e+110) || ~((y <= 2.15e+171)))
		tmp = y * (t / a);
	else
		tmp = t + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -1e+110], N[Not[LessEqual[y, 2.15e+171]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e110 or 2.15000000000000004e171 < y

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 41.7%

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

   if -1e110 < y < 2.15000000000000004e171

   1. Initial program 84.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{t + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+110} \lor \neg \left(y \leq 2.15 \cdot 10^{+171}\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+110}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+171}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1e+110) (* y (/ t a)) (if (<= y 2.15e+171) (+ t x) (* (/ y a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1e+110) {
		tmp = y * (t / a);
	} else if (y <= 2.15e+171) {
		tmp = t + x;
	} else {
		tmp = (y / 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 (y <= (-1d+110)) then
        tmp = y * (t / a)
    else if (y <= 2.15d+171) then
        tmp = t + x
    else
        tmp = (y / 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 (y <= -1e+110) {
		tmp = y * (t / a);
	} else if (y <= 2.15e+171) {
		tmp = t + x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1e+110:
		tmp = y * (t / a)
	elif y <= 2.15e+171:
		tmp = t + x
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1e+110)
		tmp = Float64(y * Float64(t / a));
	elseif (y <= 2.15e+171)
		tmp = Float64(t + x);
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1e+110)
		tmp = y * (t / a);
	elseif (y <= 2.15e+171)
		tmp = t + x;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1e+110], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+171], N[(t + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e110

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 41.6%

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

   if -1e110 < y < 2.15000000000000004e171

   1. Initial program 84.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 2.15000000000000004e171 < y

   1. Initial program 81.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 42.2%

      \[\leadsto \frac{y}{a} \cdot t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+37} \lor \neg \left(z \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.5e+37) (not (<= z 1.3e-44))) (+ t x) (* (- x) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.5e+37) || !(z <= 1.3e-44)) {
		tmp = t + x;
	} else {
		tmp = -x * -1.0;
	}
	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 <= (-4.5d+37)) .or. (.not. (z <= 1.3d-44))) then
        tmp = t + x
    else
        tmp = -x * (-1.0d0)
    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 <= -4.5e+37) || !(z <= 1.3e-44)) {
		tmp = t + x;
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.5e+37) or not (z <= 1.3e-44):
		tmp = t + x
	else:
		tmp = -x * -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.5e+37) || !(z <= 1.3e-44))
		tmp = Float64(t + x);
	else
		tmp = Float64(Float64(-x) * -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.5e+37) || ~((z <= 1.3e-44)))
		tmp = t + x;
	else
		tmp = -x * -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.5e+37], N[Not[LessEqual[z, 1.3e-44]], $MachinePrecision]], N[(t + x), $MachinePrecision], N[((-x) * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.49999999999999962e37 or 1.2999999999999999e-44 < z

   1. Initial program 73.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-+.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if -4.49999999999999962e37 < z < 1.2999999999999999e-44

   1. Initial program 96.9%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

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

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.6%

      \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+37} \lor \neg \left(z \leq 1.3 \cdot 10^{-44}\right):\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in z around inf

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

   1. lower-+.f64 57.2

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

5. Applied rewrites 57.2%

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

Developer Target 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - z}{a - z} \cdot t\\ \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y z) (- a z)) t))))
   (if (< t -1.0682974490174067e-39)
     t_1
     (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} 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 - z) / (a - z)) * t)
    if (t < (-1.0682974490174067d-39)) then
        tmp = t_1
    else if (t < 3.9110949887586375d-141) then
        tmp = x + (((y - z) * t) / (a - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) / (a - z)) * t)
	tmp = 0
	if t < -1.0682974490174067e-39:
		tmp = t_1
	elif t < 3.9110949887586375e-141:
		tmp = x + (((y - z) * t) / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * t))
	tmp = 0.0
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) / (a - z)) * t);
	tmp = 0.0;
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = x + (((y - z) * t) / (a - z));
	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 - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.0682974490174067e-39], t$95$1, If[Less[t, 3.9110949887586375e-141], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025008 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -10682974490174067/10000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 312887599100691/80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t)))))

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